Calculus: Early Transcendentals (2nd Edition)
Calculus: Early Transcendentals (2nd Edition)
2nd Edition
ISBN: 9780321947345
Author: William L. Briggs, Lyle Cochran, Bernard Gillett
Publisher: PEARSON
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Numerical Integration
In a mathematical investigation, numerical integration comprises a wide group of calculations for computing the mathematical estimation of definite integral, and likewise, the term is moreover in some cases used to depict the numerical solution of differ…
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Chapter 6.8, Problem 32E

Integrals with general bases Evaluate the following integrals.

32. ∫ 4 cot x sin 2 x d x

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Chapter 6 Solutions

Calculus: Early Transcendentals (2nd Edition)

Ch. 6.1 - Explain the meaning of position, displacement, and...Ch. 6.1 - Suppose the velocity of an object moving along a...Ch. 6.1 - Given the velocity function v of an object moving...Ch. 6.1 - Explain how to use definite integrals to find the...Ch. 6.1 - Prob. 5ECh. 6.1 - What is the result of integrating a population...Ch. 6.1 - Displacement and distance from velocity Consider...Ch. 6.1 - Prob. 8ECh. 6.1 - Displacement from velocity Assume t is time...Ch. 6.1 - Displacement from velocity Assume t is time...
Ch. 6.1 - Displacement from velocity Assume t is time...Ch. 6.1 - Displacement from velocity Assume t is time...Ch. 6.1 - Prob. 13ECh. 6.1 - Displacement from velocity Assume t is time...Ch. 6.1 - Position from velocity Consider an object moving...Ch. 6.1 - Prob. 16ECh. 6.1 - Prob. 17ECh. 6.1 - Prob. 18ECh. 6.1 - Prob. 19ECh. 6.1 - Prob. 20ECh. 6.1 - Oscillating motion A mass hanging from a spring is...Ch. 6.1 - Cycling distance A cyclist rides down a long...Ch. 6.1 - Flying into a headwind The velocity (in mi/hr) of...Ch. 6.1 - Day hike The velocity (in mi/hr) of a hiker...Ch. 6.1 - Piecewise velocity The velocity of a (fast)...Ch. 6.1 - Probe speed A data collection probe is dropped...Ch. 6.1 - Position and velocity from acceleration Find the...Ch. 6.1 - Prob. 28ECh. 6.1 - Position and velocity from acceleration Find the...Ch. 6.1 - Prob. 30ECh. 6.1 - Prob. 31ECh. 6.1 - Prob. 32ECh. 6.1 - Prob. 33ECh. 6.1 - Prob. 34ECh. 6.1 - Prob. 35ECh. 6.1 - Prob. 36ECh. 6.1 - Approaching a station At t = 0, a train...Ch. 6.1 - Prob. 38ECh. 6.1 - Oil production An oil refinery produces oil at a...Ch. 6.1 - Population growth 40. Starting with an initial...Ch. 6.1 - Population growth 41. When records were first kept...Ch. 6.1 - Population growth 42. The population of a...Ch. 6.1 - Population growth 43. A culture of bacteria in a...Ch. 6.1 - Flow rates in the Spokane River The daily...Ch. 6.1 - Marginal cost Consider the following marginal cost...Ch. 6.1 - Marginal cost Consider the following marginal cost...Ch. 6.1 - Marginal cost Consider the following marginal cost...Ch. 6.1 - Prob. 48ECh. 6.1 - Explain why or why not Determine whether the...Ch. 6.1 - Velocity graphs The figures show velocity...Ch. 6.1 - Velocity graphs The figures show velocity...Ch. 6.1 - Equivalent constant velocity Consider the...Ch. 6.1 - Equivalent constant velocity Consider the...Ch. 6.1 - Equivalent constant velocity Consider the...Ch. 6.1 - Equivalent constant velocity Consider the...Ch. 6.1 - Where do they meet? Kelly started at noon (t = 0)...Ch. 6.1 - Prob. 57ECh. 6.1 - Two runners At noon (t = 0), Alicia starts running...Ch. 6.1 - Prob. 59ECh. 6.1 - Filling a tank A 2000-liter cistern is empty when...Ch. 6.1 - Depletion of natural resources Suppose that r(t) =...Ch. 6.1 - Snowplow problem With snow on the ground and...Ch. 6.1 - Filling a reservoir A reservoir with a capacity of...Ch. 6.1 - Blood flow A typical human heart pumps 70 mL of...Ch. 6.1 - Prob. 65ECh. 6.1 - Oscillating growth rates Some species have growth...Ch. 6.1 - Power and energy Power and energy are often used...Ch. 6.1 - Variable gravity At Earths surface, the...Ch. 6.1 - Another look at the Fundamental Theorem 69....Ch. 6.1 - Another look at the Fundamental Theorem 70. Use...Ch. 6.1 - Another look at the Fundamental Theorem 71. Use...Ch. 6.1 - Another look at the Fundamental Theorem 72....Ch. 6.2 - Draw the graphs of two functions f and g that are...Ch. 6.2 - Prob. 2ECh. 6.2 - Make a sketch to show a case in which the area...Ch. 6.2 - Make a sketch to show a case in which the area...Ch. 6.2 - Finding area Determine the area of the shaded...Ch. 6.2 - Finding area Determine the area of the shaded...Ch. 6.2 - Finding area Determine the area of the shaded...Ch. 6.2 - Finding area Determine the area of the shaded...Ch. 6.2 - Regions between curves Sketch the region and find...Ch. 6.2 - Prob. 10ECh. 6.2 - Regions between curves Sketch the region and find...Ch. 6.2 - Regions between curves Sketch the region and find...Ch. 6.2 - Prob. 13ECh. 6.2 - Regions between curves Sketch the region and find...Ch. 6.2 - Prob. 15ECh. 6.2 - Prob. 16ECh. 6.2 - Prob. 17ECh. 6.2 - Prob. 18ECh. 6.2 - Compound regions Sketch each region (if a figure...Ch. 6.2 - Compound regions Sketch each region (if a figure...Ch. 6.2 - Compound regions Sketch each region (if a figure...Ch. 6.2 - Compound regions Sketch each region (if a figure...Ch. 6.2 - Integrating with respect to y Determine the area...Ch. 6.2 - Prob. 24ECh. 6.2 - Prob. 25ECh. 6.2 - Prob. 26ECh. 6.2 - Prob. 27ECh. 6.2 - Two approaches Express the area of the following...Ch. 6.2 - Prob. 29ECh. 6.2 - Two approaches Express the area of the following...Ch. 6.2 - Two approaches Find the area of the following...Ch. 6.2 - Prob. 32ECh. 6.2 - Any method Use any method (including geometry) to...Ch. 6.2 - Prob. 34ECh. 6.2 - Prob. 35ECh. 6.2 - Any method Use any method (including geometry) to...Ch. 6.2 - Any method Use any method (including geometry) to...Ch. 6.2 - Any method Use any method (including geometry) to...Ch. 6.2 - Explain why or why not Determine whether the...Ch. 6.2 - Regions between curves Sketch the region and find...Ch. 6.2 - Prob. 41ECh. 6.2 - Prob. 42ECh. 6.2 - Prob. 43ECh. 6.2 - Prob. 44ECh. 6.2 - Prob. 45ECh. 6.2 - Prob. 46ECh. 6.2 - Prob. 47ECh. 6.2 - Either method Use the most efficient strategy for...Ch. 6.2 - Prob. 49ECh. 6.2 - Prob. 50ECh. 6.2 - Comparing areas Let f(x) = xp and g(x) = x1/q,...Ch. 6.2 - Complicated regions Find the area of the regions...Ch. 6.2 - Complicated regions Find the area of the regions...Ch. 6.2 - Complicated regions Find the area of the regions...Ch. 6.2 - Prob. 55ECh. 6.2 - Prob. 56ECh. 6.2 - Prob. 57ECh. 6.2 - Prob. 58ECh. 6.2 - Prob. 59ECh. 6.2 - Prob. 60ECh. 6.2 - Bisecting regions For each region R, find the...Ch. 6.2 - Prob. 62ECh. 6.2 - Prob. 63ECh. 6.2 - Geometric probability Suppose a dartboard occupies...Ch. 6.2 - Lorenz curves and the Gini index A Lorenz curve is...Ch. 6.2 - Equal area properties for parabolas Consider the...Ch. 6.2 - Minimum area Graph the curves y = (x + 1)(x 2)...Ch. 6.2 - Prob. 68ECh. 6.2 - Area of a curve defined implicitly Determine the...Ch. 6.2 - Prob. 70ECh. 6.2 - Area function for a cubic Consider the cubic...Ch. 6.2 - Differences of even functions Assume f and g are...Ch. 6.2 - Prob. 73ECh. 6.2 - Shifting sines Consider the functions f(x) = a sin...Ch. 6.3 - Suppose a cut is made through a solid object...Ch. 6.3 - A solid has a circular base and cross sections...Ch. 6.3 - Prob. 3ECh. 6.3 - Prob. 4ECh. 6.3 - Why is the disk method a special case of the...Ch. 6.3 - Prob. 6ECh. 6.3 - General slicing method Use the general slicing...Ch. 6.3 - General slicing method Use the general slicing...Ch. 6.3 - General slicing method Use the general slicing...Ch. 6.3 - Prob. 10ECh. 6.3 - General slicing method Use the general slicing...Ch. 6.3 - Prob. 12ECh. 6.3 - General slicing method Use the general slicing...Ch. 6.3 - Prob. 14ECh. 6.3 - Prob. 15ECh. 6.3 - Prob. 16ECh. 6.3 - Disk method Let R be the region bounded by the...Ch. 6.3 - Disk method Let R be the region bounded by the...Ch. 6.3 - Disk method Let R be the region bounded by the...Ch. 6.3 - Disk method Let R be the region bounded by the...Ch. 6.3 - Disk method Let R be the region bounded by the...Ch. 6.3 - Disk method Let R be the region bounded by the...Ch. 6.3 - Prob. 23ECh. 6.3 - Prob. 24ECh. 6.3 - Prob. 25ECh. 6.3 - Prob. 26ECh. 6.3 - Washer method Let R be the region bounded by the...Ch. 6.3 - Washer method Let R be the region bounded by the...Ch. 6.3 - Washer method Let R be the region bounded by the...Ch. 6.3 - Washer method Let R be the region bounded by the...Ch. 6.3 - Prob. 31ECh. 6.3 - Washer method Let R be the region bounded by the...Ch. 6.3 - Washer method Let R be the region bounded by the...Ch. 6.3 - Washer method Let R be the region bounded by the...Ch. 6.3 - Disks/washers about the y-axis Let R be the region...Ch. 6.3 - Disks/washers about the y-axis Let R be the region...Ch. 6.3 - Prob. 37ECh. 6.3 - Disks/washers about the y-axis Let R be the region...Ch. 6.3 - Disks/washers about the y-axis Let R be the region...Ch. 6.3 - Disks/washers about the y-axis Let R be the region...Ch. 6.3 - Which is greater? For the following regions R,...Ch. 6.3 - Prob. 42ECh. 6.3 - Prob. 43ECh. 6.3 - Prob. 44ECh. 6.3 - Revolution about other axes Find the volume of the...Ch. 6.3 - Revolution about other axes Find the volume of the...Ch. 6.3 - Revolution about other axes Find the volume of the...Ch. 6.3 - Revolution about other axes Find the volume of the...Ch. 6.3 - Revolution about other axes Find the volume of the...Ch. 6.3 - Revolution about other axes Find the volume of the...Ch. 6.3 - Revolution about other axes Find the volume of the...Ch. 6.3 - Prob. 52ECh. 6.3 - Explain why or why not Determine whether the...Ch. 6.3 - Prob. 54ECh. 6.3 - Prob. 55ECh. 6.3 - Prob. 56ECh. 6.3 - Prob. 57ECh. 6.3 - Prob. 58ECh. 6.3 - Prob. 59ECh. 6.3 - Prob. 60ECh. 6.3 - Fermats volume calculation (1636) Let R be the...Ch. 6.3 - Solid from a piecewise function Let...Ch. 6.3 - Prob. 63ECh. 6.3 - Volume of a wooden object A solid wooden object...Ch. 6.3 - Cylinder, cone, hemisphere A right circular...Ch. 6.3 - Water in a bowl A hemispherical bowl of radius 8...Ch. 6.3 - A torus (doughnut) Find the volume of the torus...Ch. 6.3 - Which is greater? Let R be the region bounded by y...Ch. 6.3 - Prob. 69ECh. 6.3 - Prob. 70ECh. 6.4 - Assume f and g are continuous with f(x) g(x) on...Ch. 6.4 - Fill in the blanks: A region R is revolved about...Ch. 6.4 - Fill in the blanks: A region R is revolved about...Ch. 6.4 - Prob. 4ECh. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Prob. 8ECh. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Prob. 11ECh. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Prob. 16ECh. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Shell method Let R be the region bounded by the...Ch. 6.4 - Shell method Use the shell method to find the...Ch. 6.4 - Shell method Use the shell method to find the...Ch. 6.4 - Shell method Use the shell method to find the...Ch. 6.4 - Shell method Use the shell method to find the...Ch. 6.4 - Prob. 31ECh. 6.4 - Shell method Use the shell method to find the...Ch. 6.4 - Shell method about other lines Let R be the region...Ch. 6.4 - Shell method about other lines Let R be the region...Ch. 6.4 - Shell method about other lines Let R be the region...Ch. 6.4 - Shell method about other lines Let R be the region...Ch. 6.4 - Different axes of revolution Use either the washer...Ch. 6.4 - Different axes of revolution Use either the washer...Ch. 6.4 - Different axes of revolution Use either the washer...Ch. 6.4 - Different axes of revolution Use either the washer...Ch. 6.4 - Washers vs. shells Let R be the region bounded by...Ch. 6.4 - Prob. 42ECh. 6.4 - Prob. 43ECh. 6.4 - Washers vs. shells Let R be the region bounded by...Ch. 6.4 - Prob. 45ECh. 6.4 - Prob. 46ECh. 6.4 - Washers vs. shells Let R be the region bounded by...Ch. 6.4 - Prob. 48ECh. 6.4 - Explain why or why not Determine whether the...Ch. 6.4 - Prob. 50ECh. 6.4 - Prob. 51ECh. 6.4 - Prob. 52ECh. 6.4 - Prob. 53ECh. 6.4 - Prob. 54ECh. 6.4 - Choose your method Find the volume of the...Ch. 6.4 - Choose your method Find the volume of the...Ch. 6.4 - Choose your method Find the volume of the...Ch. 6.4 - Prob. 58ECh. 6.4 - Choose your method Find the volume of the...Ch. 6.4 - Choose your method Find the volume of the...Ch. 6.4 - Choose your method Find the volume of the...Ch. 6.4 - The solid formed when the region bounded by y=x,...Ch. 6.4 - Prob. 63ECh. 6.4 - A hemisphere by three methods Let R be the region...Ch. 6.4 - Prob. 65ECh. 6.4 - A spherical cap by three methods Consider the cap...Ch. 6.4 - Prob. 67ECh. 6.4 - Wedge from a tree Imagine a cylindrical tree of...Ch. 6.4 - Prob. 69ECh. 6.4 - Prob. 70ECh. 6.4 - Prob. 71ECh. 6.4 - Ellipsoids An ellipse centered at the origin is...Ch. 6.4 - Change of variables Suppose f(x) 0 for all x and...Ch. 6.4 - Equal integrals Without evaluating integrals,...Ch. 6.4 - Volumes without calculus Solve the following...Ch. 6.5 - Explain the steps required to find the length of a...Ch. 6.5 - Explain the steps required to find the length of a...Ch. 6.5 - Setting up arc length integrals Write and...Ch. 6.5 - Setting up arc length integrals Write and...Ch. 6.5 - Setting up arc length integrals Write and...Ch. 6.5 - Setting up arc length integrals Write and...Ch. 6.5 - Arc length calculations Find the arc length of the...Ch. 6.5 - Arc length calculations Find the arc length of the...Ch. 6.5 - Arc lezngth calculations Find the arc length of...Ch. 6.5 - Arc length calculations Find the arc length of the...Ch. 6.5 - Arc length calculations Find the arc length of the...Ch. 6.5 - Arc length calculations Find the arc length of the...Ch. 6.5 - Arc length calculations Find the arc length of the...Ch. 6.5 - Arc length calculations Find the arc length of the...Ch. 6.5 - Prob. 15ECh. 6.5 - Arc length calculations Find the arc length of the...Ch. 6.5 - Arc length by calculator a. Write and simplify the...Ch. 6.5 - Arc length by calculator a. Write and simplify the...Ch. 6.5 - Arc length by calculator a. Write and simplify the...Ch. 6.5 - Arc length by calculator a. Write and simplify the...Ch. 6.5 - Arc length by calculator a. Write and simplify the...Ch. 6.5 - Arc length by calculator a. Write and simplify the...Ch. 6.5 - Arc length by calculator a. Write and simplify the...Ch. 6.5 - Arc length by calculator a. Write and simplify the...Ch. 6.5 - Arc length by calculator a. Write and simplify the...Ch. 6.5 - Arc length by calculator a. Write and simplify the...Ch. 6.5 - Arc length calculations with respect to y Find the...Ch. 6.5 - Arc length calculations with respect to y Find the...Ch. 6.5 - Prob. 29ECh. 6.5 - Arc length calculations with respect to y Find the...Ch. 6.5 - Explain why or why not Determine whether the...Ch. 6.5 - Arc length for a line Consider the segment of the...Ch. 6.5 - Functions from arc length What differentiable...Ch. 6.5 - Function from arc length Find a curve that passes...Ch. 6.5 - Prob. 35ECh. 6.5 - Prob. 36ECh. 6.5 - Golden Gate cables The profile of the cables on a...Ch. 6.5 - Gateway Arch The shape of the Gateway Arch in St....Ch. 6.5 - Lengths of related curves Suppose the graph of f...Ch. 6.5 - Prob. 40ECh. 6.5 - A family of exponential functions a. Show that the...Ch. 6.5 - Bernoullis parabolas Johann Bernoulli (16671748)...Ch. 6.6 - What is the area of the curved surface of a right...Ch. 6.6 - A frustum of a cone is generated by revolving the...Ch. 6.6 - Suppose f is positive and differentiable on [a,...Ch. 6.6 - Suppose g is positive and differentiable on [c,...Ch. 6.6 - Computing surface areas Find the area of the...Ch. 6.6 - Computing surface areas Find the area of the...Ch. 6.6 - Computing surface areas Find the area of the...Ch. 6.6 - Computing surface areas Find the area of the...Ch. 6.6 - Prob. 9ECh. 6.6 - Computing surface areas Find the area of the...Ch. 6.6 - Computing surface areas Find the area of the...Ch. 6.6 - Prob. 12ECh. 6.6 - Prob. 13ECh. 6.6 - Computing surface areas Find the area of the...Ch. 6.6 - Painting surfaces A 1.5-mm layer of paint is...Ch. 6.6 - Painting surfaces A 1.5-mm layer of paint is...Ch. 6.6 - Revolving about the y-axis Find the area of the...Ch. 6.6 - Revolving about the y-axis Find the area of the...Ch. 6.6 - Revolving about the y-axis Find the area of the...Ch. 6.6 - Revolving about the y-axis Find the area of the...Ch. 6.6 - Explain why or why not Determine whether the...Ch. 6.6 - Surface area calculations Use the method of your...Ch. 6.6 - Surface area calculations Use the method of your...Ch. 6.6 - Surface area calculations Use the method of your...Ch. 6.6 - Prob. 25ECh. 6.6 - Prob. 26ECh. 6.6 - T 2629. Surface area using technology Consider the...Ch. 6.6 - Prob. 28ECh. 6.6 - Surface area using technology Consider the...Ch. 6.6 - Cones and cylinders The volume of a cone of radius...Ch. 6.6 - Prob. 31ECh. 6.6 - Surface area of a torus When the circle x2 + (y ...Ch. 6.6 - Zones of a sphere Suppose a sphere of radius r is...Ch. 6.6 - Prob. 34ECh. 6.6 - Surface-area-to-volume ratio (SAV) In the design...Ch. 6.6 - Surface area of a frustum Show that the surface...Ch. 6.6 - Scaling surface area Let f be a nonnegative...Ch. 6.6 - Surface plus cylinder Suppose f is a nonnegative...Ch. 6.7 - Suppose a 1-m cylindrical bar has a constant...Ch. 6.7 - Explain how to find the mass of a one-dimensional...Ch. 6.7 - How much work is required to move an object from x...Ch. 6.7 - Why is integration used to find the work done by a...Ch. 6.7 - Why is integration used to find the work required...Ch. 6.7 - Why is integration used to find the total force on...Ch. 6.7 - What is the pressure on a horizontal surface with...Ch. 6.7 - Explain why you integrate in the vertical...Ch. 6.7 - Mass of one-dimensional objects Find the mass of...Ch. 6.7 - Mass of one-dimensional objects Find the mass of...Ch. 6.7 - Mass of one-dimensional objects Find the mass of...Ch. 6.7 - Mass of one-dimensional objects Find the mass of...Ch. 6.7 - Mass of one-dimensional objects Find the mass of...Ch. 6.7 - Mass of one-dimensional objects Find the mass of...Ch. 6.7 - Mass of one-dimensional objects Find the mass of...Ch. 6.7 - Mass of one-dimensional objects Find the mass of...Ch. 6.7 - Work from force How much work is required to move...Ch. 6.7 - Work from force How much work is required to move...Ch. 6.7 - Compressing and stretching a spring Suppose a...Ch. 6.7 - Compressing and stretching a spring Suppose a...Ch. 6.7 - Work done by a spring A spring on a horizontal...Ch. 6.7 - Shock absorber A heavy-duty shock absorber is...Ch. 6.7 - Calculating work for different springs Calculate...Ch. 6.7 - Calculating work for different springs Calculate...Ch. 6.7 - Calculating work for different springs Calculate...Ch. 6.7 - Work function A spring has a restoring force given...Ch. 6.7 - Emptying a swimming pool A swimming pool has the...Ch. 6.7 - Emptying a cylindrical tank A cylindrical water...Ch. 6.7 - Emptying a half-full cylindrical tank Suppose the...Ch. 6.7 - Emptying a partially filled swimming pool If the...Ch. 6.7 - Emptying a conical tank A water tank is shaped...Ch. 6.7 - Emptying a real swimming pool A swimming pool is...Ch. 6.7 - Prob. 33ECh. 6.7 - Emptying a water trough A water trough has a...Ch. 6.7 - Emptying a water trough A cattle trough has a...Ch. 6.7 - Prob. 36ECh. 6.7 - Emptying a conical tank An inverted cone is 2 m...Ch. 6.7 - Force on dams The following figures show the shape...Ch. 6.7 - Force on dams The following figures show the shape...Ch. 6.7 - Force on dams The following figures show the shape...Ch. 6.7 - Force on dams The following figures show the shape...Ch. 6.7 - Parabolic dam The lower edge of a dam is defined...Ch. 6.7 - Prob. 43ECh. 6.7 - Force on the end of a tank Determine the force on...Ch. 6.7 - Force on a building A large building shaped like a...Ch. 6.7 - Force on a window A diving pool that is 4 m deep...Ch. 6.7 - Force on a window A diving pool that is 4 m deep...Ch. 6.7 - Force on a window A diving pool that is 4 m deep...Ch. 6.7 - Explain why or why not Determine whether the...Ch. 6.7 - Prob. 50ECh. 6.7 - A nonlinear spring Hookes law is applicable to...Ch. 6.7 - Prob. 52ECh. 6.7 - Drinking juice A glass has circular cross sections...Ch. 6.7 - Upper and lower half A cylinder with height 8 m...Ch. 6.7 - Work in a gravitational field For large distances...Ch. 6.7 - Prob. 56ECh. 6.7 - Winding a chain A 30-m-long chain hangs vertically...Ch. 6.7 - Coiling a rope A 60-m-long, 9.4-mm-diameter rope...Ch. 6.7 - Lifting a pendulum A body of mass m is suspended...Ch. 6.7 - Prob. 60ECh. 6.7 - Prob. 61ECh. 6.7 - Prob. 62ECh. 6.7 - Critical depth A large tank has a plastic window...Ch. 6.7 - Buoyancy Archimedes principle says that the...Ch. 6.8 - Prob. 1ECh. 6.8 - Prob. 2ECh. 6.8 - Evaluate 4xdx.Ch. 6.8 - What is the inverse function of ln x, and what are...Ch. 6.8 - Express 3x, x, and xsin x using the base e.Ch. 6.8 - Evaluate ddx(3x).Ch. 6.8 - Prob. 7ECh. 6.8 - Derivatives with ln x Evaluate the following...Ch. 6.8 - Derivatives with ln x Evaluate the following...Ch. 6.8 - Derivatives with ln x Evaluate the following...Ch. 6.8 - Derivatives with ln x Evaluate the following...Ch. 6.8 - Derivatives with ln x Evaluate the following...Ch. 6.8 - Integrals with ln x Evaluate the following...Ch. 6.8 - Prob. 14ECh. 6.8 - Integrals with ln x Evaluate the following...Ch. 6.8 - Integrals with ln x Evaluate the following...Ch. 6.8 - Integrals with ln x Evaluate the following...Ch. 6.8 - Integrals with ln x Evaluate the following...Ch. 6.8 - Integrals with ln x Evaluate the following...Ch. 6.8 - Integrals with ln x Evaluate the following...Ch. 6.8 - Integrals with ex Evaluate the following...Ch. 6.8 - Integrals with ex Evaluate the following...Ch. 6.8 - Integrals with ex Evaluate the following...Ch. 6.8 - Integrals with ex Evaluate the following...Ch. 6.8 - Integrals with ex Evaluate the following...Ch. 6.8 - Integrals with ex Evaluate the following...Ch. 6.8 - Integrals with general bases Evaluate the...Ch. 6.8 - Integrals with general bases Evaluate the...Ch. 6.8 - Integrals with general bases Evaluate the...Ch. 6.8 - Integrals with general bases Evaluate the...Ch. 6.8 - Integrals with general bases Evaluate the...Ch. 6.8 - Integrals with general bases Evaluate the...Ch. 6.8 - Derivatives Evaluate the derivatives of the...Ch. 6.8 - Derivatives Evaluate the derivatives of the...Ch. 6.8 - Derivatives Evaluate the derivatives of the...Ch. 6.8 - Derivatives Evaluate the derivatives of the...Ch. 6.8 - Derivatives Evaluate the derivatives of the...Ch. 6.8 - Derivatives Evaluate the derivatives of the...Ch. 6.8 - Derivatives Evaluate the derivatives of the...Ch. 6.8 - Derivatives Evaluate the derivatives of the...Ch. 6.8 - Prob. 41ECh. 6.8 - Prob. 42ECh. 6.8 - Prob. 43ECh. 6.8 - Prob. 44ECh. 6.8 - Prob. 45ECh. 6.8 - Prob. 46ECh. 6.8 - Prob. 47ECh. 6.8 - Prob. 48ECh. 6.8 - Prob. 49ECh. 6.8 - Miscellaneous derivatives Compute the following...Ch. 6.8 - Miscellaneous derivatives Compute the following...Ch. 6.8 - Prob. 52ECh. 6.8 - Miscellaneous derivatives Compute the following...Ch. 6.8 - Miscellaneous derivatives Compute the following...Ch. 6.8 - Miscellaneous derivatives Compute the following...Ch. 6.8 - Miscellaneous derivatives Compute the following...Ch. 6.8 - Miscellaneous derivatives Compute the following...Ch. 6.8 - Miscellaneous integrals Evaluate the following...Ch. 6.8 - Miscellaneous integrals Evaluate the following...Ch. 6.8 - Miscellaneous integrals Evaluate the following...Ch. 6.8 - Miscellaneous integrals Evaluate the following...Ch. 6.8 - Miscellaneous integrals Evaluate the following...Ch. 6.8 - Miscellaneous integrals Evaluate the following...Ch. 6.8 - Miscellaneous integrals Evaluate the following...Ch. 6.8 - Miscellaneous integrals Evaluate the following...Ch. 6.8 - Miscellaneous integrals Evaluate the following...Ch. 6.8 - Miscellaneous integrals Evaluate the following...Ch. 6.8 - Miscellaneous integrals Evaluate the following...Ch. 6.8 - Probability as an integral Two points P and Q are...Ch. 6.8 - Prob. 70ECh. 6.8 - Prob. 71ECh. 6.8 - Prob. 72ECh. 6.8 - Prob. 73ECh. 6.8 - Prob. 74ECh. 6.8 - Prob. 75ECh. 6.9 - In terms of relative growth rate, what is the...Ch. 6.9 - Prob. 2ECh. 6.9 - Prob. 3ECh. 6.9 - Prob. 4ECh. 6.9 - Prob. 5ECh. 6.9 - Prob. 6ECh. 6.9 - Give two examples of processes that are modeled by...Ch. 6.9 - Give two examples of processes that are modeled by...Ch. 6.9 - Absolute and relative growth rates Two functions f...Ch. 6.9 - Absolute and relative growth rates Two functions f...Ch. 6.9 - Designing exponential growth functions Devise the...Ch. 6.9 - Prob. 12ECh. 6.9 - Prob. 13ECh. 6.9 - Prob. 14ECh. 6.9 - Prob. 15ECh. 6.9 - Designing exponential growth functions Devise the...Ch. 6.9 - Projection sensitivity According to the 2010...Ch. 6.9 - Energy consumption On the first day of the year (t...Ch. 6.9 - Prob. 19ECh. 6.9 - Prob. 20ECh. 6.9 - Prob. 21ECh. 6.9 - Designing exponential decay functions Devise an...Ch. 6.9 - Designing exponential decay functions Devise an...Ch. 6.9 - Designing exponential decay functions Devise an...Ch. 6.9 - Prob. 25ECh. 6.9 - Prob. 26ECh. 6.9 - Atmospheric pressure The pressure of Earths...Ch. 6.9 - Carbon dating The half-life of C-14 is about 5730...Ch. 6.9 - Uranium dating Uranium-238 (U-238) has a half-life...Ch. 6.9 - Radioiodine treatment Roughly 12,000 Americans are...Ch. 6.9 - Explain why or why not Determine whether the...Ch. 6.9 - Tripling time A quantity increases according to...Ch. 6.9 - Constant doubling time Prove that the doubling...Ch. 6.9 - Prob. 34ECh. 6.9 - A slowing race Starting at the same time and...Ch. 6.9 - Prob. 36ECh. 6.9 - Compounded inflation The U.S. government reports...Ch. 6.9 - Acceleration, velocity, position Suppose the...Ch. 6.9 - Air resistance (adapted from Putnam Exam, 1939) An...Ch. 6.9 - A running model A model for the startup of a...Ch. 6.9 - Tumor growth Suppose the cells of a tumor are...Ch. 6.9 - Prob. 42ECh. 6.9 - Prob. 43ECh. 6.9 - Geometric means A quantity grows exponentially...Ch. 6.9 - Equivalent growth functions The same exponential...Ch. 6.9 - General relative growth rates Define the relative...Ch. 6.10 - State the definition of the hyperbolic cosine and...Ch. 6.10 - Sketch the graphs of y = cosh x, y sinh x, and y...Ch. 6.10 - What is the fundamental identity for hyperbolic...Ch. 6.10 - Prob. 4ECh. 6.10 - Express sinh1 x in terms of logarithms.Ch. 6.10 - Prob. 6ECh. 6.10 - Prob. 7ECh. 6.10 - On what interval is the formula d/dx (tanh1 x) =...Ch. 6.10 - Prob. 9ECh. 6.10 - Prob. 10ECh. 6.10 - Verifying identities Verify each identity using...Ch. 6.10 - Verifying identities Verify each identity using...Ch. 6.10 - Verifying identities Verify each identity using...Ch. 6.10 - Verifying identities Verify each identity using...Ch. 6.10 - Verifying identities Verify each identity using...Ch. 6.10 - Verifying identities Use the given identity to...Ch. 6.10 - Verifying identities Use the given identity to...Ch. 6.10 - Prob. 18ECh. 6.10 - Derivative formulas Derive the following...Ch. 6.10 - Derivative formulas Derive the following...Ch. 6.10 - Derivative formulas Derive the following...Ch. 6.10 - Derivatives Compute dy/dx for the following...Ch. 6.10 - Derivatives Compute dy/dx for the following...Ch. 6.10 - Derivatives Compute dy/dx for the following...Ch. 6.10 - Derivatives Compute dy/dx for the following...Ch. 6.10 - Derivatives Compute dy/dx for the following...Ch. 6.10 - Derivatives Compute dy/dx for the following...Ch. 6.10 - Derivatives Compute dy/dx for the following...Ch. 6.10 - Derivatives Compute dy/dx for the following...Ch. 6.10 - Prob. 30ECh. 6.10 - Indefinite integrals Determine each indefinite...Ch. 6.10 - Prob. 32ECh. 6.10 - Indefinite integrals Determine each indefinite...Ch. 6.10 - Indefinite integrals Determine each indefinite...Ch. 6.10 - Indefinite integrals Determine each indefinite...Ch. 6.10 - Indefinite integrals Determine each indefinite...Ch. 6.10 - Definite integrals Evaluate each definite...Ch. 6.10 - Prob. 38ECh. 6.10 - Definite integrals Evaluate each definite...Ch. 6.10 - Definite integrals Evaluate each definite...Ch. 6.10 - Two ways Evaluate the following integrals two...Ch. 6.10 - Two ways Evaluate the following integrals two...Ch. 6.10 - Visual approximation a. Use a graphing utility to...Ch. 6.10 - Prob. 44ECh. 6.10 - Prob. 45ECh. 6.10 - Points of intersection and area a. Sketch the...Ch. 6.10 - Derivatives Find the derivatives of the following...Ch. 6.10 - Derivatives Find the derivatives of the following...Ch. 6.10 - Derivatives Find the derivatives of the following...Ch. 6.10 - Derivatives Find the derivatives of the following...Ch. 6.10 - Prob. 51ECh. 6.10 - Prob. 52ECh. 6.10 - Indefinite integrals Determine the following...Ch. 6.10 - Prob. 54ECh. 6.10 - Indefinite integrals Determine the following...Ch. 6.10 - Prob. 56ECh. 6.10 - Indefinite integrals Determine the following...Ch. 6.10 - Prob. 58ECh. 6.10 - Prob. 59ECh. 6.10 - Prob. 60ECh. 6.10 - Prob. 61ECh. 6.10 - Prob. 62ECh. 6.10 - Prob. 63ECh. 6.10 - Prob. 64ECh. 6.10 - Catenary arch The portion of the curve y=1716coshx...Ch. 6.10 - Length of a catenary Show that the arc length of...Ch. 6.10 - Power lines A power line is attached at the same...Ch. 6.10 - Sag angle Imagine a climber clipping onto the rope...Ch. 6.10 - Wavelength The velocity of a surface wave on the...Ch. 6.10 - Wave velocity Use Exercise 69 to do the following...Ch. 6.10 - Prob. 71ECh. 6.10 - Tsunamis A tsunami is an ocean wave often caused...Ch. 6.10 - Explain why or why not Determine whether the...Ch. 6.10 - Evaluating hyperbolic functions Use a calculator...Ch. 6.10 - Evaluating hyperbolic functions Evaluate each...Ch. 6.10 - Confirming a graph The graph of f(x) = sinh x is...Ch. 6.10 - Critical points Find the critical points of the...Ch. 6.10 - Critical points a. Show that the critical points...Ch. 6.10 - Points of inflection Find the x-coordinate of the...Ch. 6.10 - Prob. 80ECh. 6.10 - Area of region Find the area of the region bounded...Ch. 6.10 - Prob. 82ECh. 6.10 - LHpital loophole Explain why lHpitals Rule fails...Ch. 6.10 - Limits Use lHpitals Rule to evaluate the following...Ch. 6.10 - Limits Use lHpitals Rule to evaluate the following...Ch. 6.10 - Prob. 86ECh. 6.10 - Prob. 87ECh. 6.10 - Prob. 88ECh. 6.10 - Additional integrals Evaluate the following...Ch. 6.10 - Additional integrals Evaluate the following...Ch. 6.10 - Prob. 91ECh. 6.10 - Additional integrals Evaluate the following...Ch. 6.10 - Kiln design Find the volume interior to the...Ch. 6.10 - Prob. 94ECh. 6.10 - Falling body When an object falling from rest...Ch. 6.10 - Prob. 96ECh. 6.10 - Prob. 97ECh. 6.10 - Prob. 98ECh. 6.10 - Prob. 99ECh. 6.10 - Prob. 100ECh. 6.10 - Prob. 101ECh. 6.10 - Prob. 102ECh. 6.10 - Prob. 103ECh. 6.10 - Prob. 104ECh. 6.10 - Prob. 105ECh. 6.10 - Prob. 106ECh. 6.10 - Prob. 107ECh. 6.10 - Prob. 108ECh. 6.10 - Arc length Use the result of Exercise 108 to find...Ch. 6.10 - Prob. 110ECh. 6.10 - Prob. 111ECh. 6.10 - Definitions of hyperbolic sine and cosine Complete...Ch. 6 - Explain why or why not Determine whether the...Ch. 6 - Displacement from velocity The velocity of an...Ch. 6 - Position, displacement, and distance A projectile...Ch. 6 - Deceleration At t = 0, a car begins decelerating...Ch. 6 - An oscillator The acceleration of an object moving...Ch. 6 - A race Starting at the same point on a straight...Ch. 6 - Fuel consumption A small plane in flight consumes...Ch. 6 - Variable flow rate Water flows out of a tank at a...Ch. 6 - Decreasing velocity A projectile is fired upward,...Ch. 6 - Decreasing velocity A projectile is fired upward,...Ch. 6 - An exponential bike ride Tom and Sue took a bike...Ch. 6 - Prob. 12RECh. 6 - Areas of regions Use any method to find the area...Ch. 6 - Prob. 14RECh. 6 - Prob. 15RECh. 6 - Prob. 16RECh. 6 - Areas of regions Use any method to find the area...Ch. 6 - Areas of regions Use any method to find the area...Ch. 6 - Areas of regions Use any method to find the area...Ch. 6 - Prob. 20RECh. 6 - Prob. 21RECh. 6 - Two methods The region R in the first quadrant...Ch. 6 - Volumes of solids Choose the general slicing...Ch. 6 - Volumes of solids Choose the general slicing...Ch. 6 - Volumes of solids Choose the general slicing...Ch. 6 - Volumes of solids Choose the general slicing...Ch. 6 - Volumes of solids Choose the general slicing...Ch. 6 - Volumes of solids Choose the general slicing...Ch. 6 - Volumes of solids Choose the general slicing...Ch. 6 - Volumes of solids Choose the general slicing...Ch. 6 - Volumes of solids Choose the general slicing...Ch. 6 - Volumes of solids Choose the general slicing...Ch. 6 - Volumes of solids Choose the general slicing...Ch. 6 - Volumes of solids Choose the general slicing...Ch. 6 - Volumes of solids Choose the general slicing...Ch. 6 - Area and volume The region R is bounded by the...Ch. 6 - Comparing volumes Let R be the region bounded by y...Ch. 6 - Multiple regions Determine the area of the region...Ch. 6 - Prob. 39RECh. 6 - Arc length Find the length of the following...Ch. 6 - Prob. 41RECh. 6 - Arc length Find the length of the following...Ch. 6 - Arc length Find the length of the following...Ch. 6 - Arc length Find the length of the following...Ch. 6 - Prob. 45RECh. 6 - Surface area and volume Let f(x)=13x3 and let R be...Ch. 6 - Surface area and volume Let f(x)=3xx2 and let R be...Ch. 6 - Surface area of a cone Find the surface area of a...Ch. 6 - Surface area and more Let f(x)=x42+116x2 and let R...Ch. 6 - Variable density in one dimension Find the mass of...Ch. 6 - Variable density in one dimension Find the mass of...Ch. 6 - Variable density in one dimension Find the mass of...Ch. 6 - Spring work a. It lakes 50 J of work to stretch a...Ch. 6 - Pumping water A cylindrical water tank has a...Ch. 6 - Force on a dam Find the total force on the face of...Ch. 6 - Integrals Evaluate the following integrals. 56....Ch. 6 - Integrals Evaluate the following integrals. 57....Ch. 6 - Integrals Evaluate the following integrals. 58....Ch. 6 - Integrals Evaluate the following integrals. 59....Ch. 6 - Integrals Evaluate the following integrals. 60....Ch. 6 - Integrals Evaluate the following integrals. 61....Ch. 6 - Integrals Evaluate the following integrals. 62....Ch. 6 - Integrals Evaluate the following integrals. 63....Ch. 6 - Radioactive decay The mass of radioactive material...Ch. 6 - Population growth Growing from an initial...Ch. 6 - Prob. 66RECh. 6 - Prob. 67RECh. 6 - Prob. 68RECh. 6 - Prob. 69RECh. 6 - Equal area property for parabolas Let f(x) = ax2 +...Ch. 6 - Derivatives of hyperbolic functions Compute the...Ch. 6 - Prob. 72RECh. 6 - Linear approximation Find the linear approximation...Ch. 6 - Limit Evaluate limx(tanhx)x.

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