Calculus: Early Transcendentals
12th Edition
ISBN: 9781119778189
Author: Anton, Howard, Bivens, Irl C., Davis, Stephen
Publisher: WILEY
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Textbook Question
Chapter 6.3, Problem 6ES
Use cylindrical shells to find the volume of the solid generated when the region enclosed by the given curves is revolved about the
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Assignment #1
Q1: Test the following series for convergence. Specify the test you use:
1
n+5
(-1)n
a) Σn=o
√n²+1
b) Σn=1 n√n+3
c) Σn=1 (2n+1)3
3n
1
d) Σn=1 3n-1
e) Σn=1
4+4n
answer problem 1a, 1b, 1c, 1d, and 1e and show work/ explain how you got the answer
Provethat
a) prove that for any irrational numbers there exists?
asequence of rational numbers Xn converg to S.
b) let S: RR be a sunctions-t.
f(x)=(x-1) arc tan (x), xe Q
3(x-1)
1+x²
x&Q
Show that lim f(x)= 0
14x
C) For any set A define the set -A=y
Chapter 6 Solutions
Calculus: Early Transcendentals
Ch. 6.1 - An integral expression for the area of the region...Ch. 6.1 - An integral expression for the area of the...Ch. 6.1 - (a) The points of intersection for the circle...Ch. 6.1 - Prob. 4QCECh. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region.Ch. 6.1 - Find the area of the shaded region by (a)...Ch. 6.1 - Find the area of the shaded region by (a)...
Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Sketch the region enclosed by the curves and find...Ch. 6.1 - Prob. 19ESCh. 6.1 - Use a graphing utility, where helpful, to find the...Ch. 6.1 - Prob. 22ESCh. 6.1 - Use a graphing utility, where helpful, to find the...Ch. 6.1 - Use a graphing utility, where helpful, to find the...Ch. 6.1 - Use a graphing utility, where helpful, to find the...Ch. 6.1 - Use a graphing utility, where helpful, to find the...Ch. 6.1 - Determine whether the statement is true or false....Ch. 6.1 - Determine whether the statement is true or false....Ch. 6.1 - Determine whether the statement is true or false....Ch. 6.1 - Determine whether the statement is true or false....Ch. 6.1 - Estimate the value of k0k1 so that the region...Ch. 6.1 - Estimate the area of the region in the first...Ch. 6.1 - Find a horizontal line y = k that divides the area...Ch. 6.1 - Find a vertical line x=k that divides the area...Ch. 6.1 - (a) Find the area of the region enclosed by the...Ch. 6.1 - Find the area between the curve y=sinx and the...Ch. 6.1 - Use Newton’s Method (Section 4.7 ), where...Ch. 6.1 - Use Newton’s Method (Section 4.7 ), where...Ch. 6.1 - Use Newton’s Method (Section 4.7 ), where...Ch. 6.1 - Find the area of the region that is enclosed by...Ch. 6.1 - Referring to the accompanying figure, use a CAS to...Ch. 6.1 - Two racers in adjacent lanes move with velocity...Ch. 6.1 - The accompanying figure shows acceleration versus...Ch. 6.1 - The accompanying figure shows the rate at which...Ch. 6.1 - Find the area of the region enclosed between the...Ch. 6.1 - Show that the area of the ellipse in the...Ch. 6.1 - Suppose that f and g are continuous on a,b but...Ch. 6.2 - A solid S extends along the x-axis from x=1 to x=3...Ch. 6.2 - A solid S is generated by revolving the region...Ch. 6.2 - A solid S is generated by revolving the region...Ch. 6.2 - A solid S is generated by revolving the region...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Prob. 16ESCh. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid whose base is the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - True-False Determine whether the statement is true...Ch. 6.2 - True-False Determine whether the statement is true...Ch. 6.2 - True-False Determine whether the statement is true...Ch. 6.2 - True-False Determine whether the statement is true...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Let V be the volume of the solid that results when...Ch. 6.2 - Find the volume of the solid generated when the...Ch. 6.2 - Find the volume of the solid generated when the...Ch. 6.2 - Consider the solid generated by revolving the...Ch. 6.2 - Consider the solid generated by revolving the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - Find the volume of the solid that results when the...Ch. 6.2 - A nose cone for a space reentry vehicle is...Ch. 6.2 - Find the volume of the solid whose base is the...Ch. 6.2 - In parts (a)-(c) find the volume of the solid...Ch. 6.2 - Prob. 50ESCh. 6.2 - Use a CAS to estimate the volume of the solid that...Ch. 6.2 - Use a CAS to estimate the volume of the solid that...Ch. 6.2 - Use a CAS to estimate the volume of the solid that...Ch. 6.2 - Use a CAS to estimate the volume of the solid that...Ch. 6.2 - The accompanying figure shows a spherical cap of...Ch. 6.2 - If fluid enters a hemispherical bowl with a radius...Ch. 6.2 - The accompanying figure shows the dimensions of a...Ch. 6.2 - Use the result in Exercise 55 to find the volume...Ch. 6.2 - As shown in the accompanying figure, a cocktail...Ch. 6.2 - Find the volume of the torus that results when the...Ch. 6.2 - A wedge is cut from a right circular cylinder of...Ch. 6.2 - Find the volume of the wedge described in Exercise...Ch. 6.2 - Two right circular cylinders of radius r have axes...Ch. 6.2 - In 1635 Bonaventura Cavalieri, a student of...Ch. 6.3 - Let R be the region between the x-axis and the...Ch. 6.3 - Let R be the region described in Quick Check...Ch. 6.3 - A solid S is generated by revolving the region...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - True-False Determine whether the statement is true...Ch. 6.3 - True-False Determine whether the statement is true...Ch. 6.3 - True-False Determine whether the statement is true...Ch. 6.3 - True-False Determine whether the statement is true...Ch. 6.3 - Use a CAS to find the volume of the solid...Ch. 6.3 - Use a CAS to find the volume of the solid...Ch. 6.3 - Consider the region to the right of the y-axis ,...Ch. 6.3 - Let K1 and R2 be regions of the form shown in the...Ch. 6.3 - (a) Use cylindrical shells to find the volume of...Ch. 6.3 - Let f be continuous and nonnegative on a,b , and...Ch. 6.3 - Using the method of cylindrical shells, set up but...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - As shown in the accompanying figure, a cylindrical...Ch. 6.3 - Use cylindrical shells to find the volume of the...Ch. 6.3 - Let Vx and Vy be the volumes of the solids that...Ch. 6.3 - (a) Find the volume V of the solid generated when...Ch. 6.4 - A function f is smooth on a,b if f is on a,b .Ch. 6.4 - If a function f is smooth on a,b , then the length...Ch. 6.4 - The distance between points 1,0 and e,1 is .Ch. 6.4 - Let L be the length of the curve y=lnx from 1,0 to...Ch. 6.4 - Use the Theorem of Pythagoras to find the length...Ch. 6.4 - Find the exact arc length of the curve over the...Ch. 6.4 - Find the exact arc length of the curve over the...Ch. 6.4 - Find the exact arc length of the curve over the...Ch. 6.4 - Find the exact arc length of the curve over the...Ch. 6.4 - Find the exact arc length of the curve over the...Ch. 6.4 - True-False Determine whether the statement is true...Ch. 6.4 - True-False Determine whether the statement is true...Ch. 6.4 - True-False Determine whether the statement is true...Ch. 6.4 - True-False Determine whether the statement is true...Ch. 6.4 - Express the exact arc length of the curve over the...Ch. 6.4 - Express the exact arc length of the curve over the...Ch. 6.4 - Consider the curve y=x2/3 . (a) Sketch the portion...Ch. 6.4 - Consider the curve segments y=x2 from x=12 to x=2...Ch. 6.4 - Follow the directions of Exercise 17 for the curve...Ch. 6.4 - Follow the directions of Exercise 17 for the curve...Ch. 6.4 - Let y=fx be a smooth curve on the closed interval...Ch. 6.4 - Use the result of Exercise 20 to show that the arc...Ch. 6.4 - A basketball player makes a successful shot from...Ch. 6.4 - Prob. 23ESCh. 6.4 - As shown in the accompanying figure, a horizontal...Ch. 6.4 - These exercises assume familiarity with the basic...Ch. 6.4 - Use the arc length formula from Exercise 26 to...Ch. 6.4 - Use the arc length formula from Exercise 26 to...Ch. 6.4 - Use the arc length formula from Exercise 26 to...Ch. 6.4 - Use the arc length formula from Exercise 26 to...Ch. 6.4 - Use the arc length formula from Exercise 26 to...Ch. 6.4 - Use the arc length formula from Exercise 26 to...Ch. 6.4 - (a) Show that the total arc length of the ellipse...Ch. 6.4 - Show that the total arc length of the ellipse...Ch. 6.4 - Writing In our discussion of Arc Length Problem...Ch. 6.5 - If f is a smooth, nonnegative function on a,b ,...Ch. 6.5 - The lateral area of the frustum with slant height...Ch. 6.5 - Anintegral expression for the area of the surface...Ch. 6.5 - An integral expression for the area of the surface...Ch. 6.5 - Find the area of the surface generated by...Ch. 6.5 - Prob. 2ESCh. 6.5 - Find the area of the surface generated by...Ch. 6.5 - Find the area of the surface generated by...Ch. 6.5 - Find the area of the surface generated by...Ch. 6.5 - Find the area of the surface generated by...Ch. 6.5 - Find the area of the surface generated by...Ch. 6.5 - Find the area of the surface generated by...Ch. 6.5 - Use a CAS to find the exact area of the surface...Ch. 6.5 - Use a CAS to find the exact area of the surface...Ch. 6.5 - Use a CAS to find the exact area of the surface...Ch. 6.5 - Prob. 12ESCh. 6.5 - Use a CAS or a calculating utility with a...Ch. 6.5 - Prob. 14ESCh. 6.5 - Use a CAS or a calculating utility with a...Ch. 6.5 - Prob. 16ESCh. 6.5 - True-False Determine whether the statement is true...Ch. 6.5 - True-False Determine whether the statement is true...Ch. 6.5 - True-False Determine whether the statement is true...Ch. 6.5 - True-False Determine whether the statement is true...Ch. 6.5 - Approximate the area of the surface using Formula...Ch. 6.5 - Approximate the area of the surface using Formula...Ch. 6.5 - Assume that y=fx is a smooth curve on the interval...Ch. 6.5 - Would it be circular reasoning to use Definition...Ch. 6.5 - Show that the area of the surface of a sphere of...Ch. 6.5 - The accompanying figure shows a spherical cap of...Ch. 6.5 - The portion of a sphere that is cut by two...Ch. 6.5 - Let y=fx be a smooth curve on the interval a,b and...Ch. 6.5 - Prob. 29ESCh. 6.5 - Let y=fx be a smooth curve on a,b and assume that...Ch. 6.6 - If a constant force of 5lb moves an object 10ft ,...Ch. 6.6 - A newton-meter is also called a . A...Ch. 6.6 - Suppose that an object moves in the positive...Ch. 6.6 - A force Fx=102xN applied in the positive x...Ch. 6.6 - Prob. 1ESCh. 6.6 - Prob. 2ESCh. 6.6 - A variable force Fx in Exercise 2 , consider the...Ch. 6.6 - Suppose that a variable force Fx is applied in the...Ch. 6.6 - A constant force of 10lb in the positive x-...Ch. 6.6 - A spring exerts a force of 6N when it is stretched...Ch. 6.6 - A spring exerts a force of 100N when it is...Ch. 6.6 - A spring whose natural length is 15cm exerts a...Ch. 6.6 - Assume that 10ftlb of work is required to stretch...Ch. 6.6 - True-False Determine whether the statement is true...Ch. 6.6 - True-False Determine whether the statement is true...Ch. 6.6 - True-False Determine whether the statement is true...Ch. 6.6 - True-False Determine whether the statement is true...Ch. 6.6 - A cylindrical tank of radius 5ft and height 9ft is...Ch. 6.6 - Solve Exercise 14 assuming that the tank is...Ch. 6.6 - A cone-shaped water reservoir is 20ft in diameter...Ch. 6.6 - The vat shown in the accompanying figure contains...Ch. 6.6 - The cylindrical tank shown in the accompanying...Ch. 6.6 - A swimming pool is built in the shape of a...Ch. 6.6 - How much work is required to fill the swimming...Ch. 6.6 - A 100ft length of steel chain weighing 15lb/ft is...Ch. 6.6 - A 3lb bucket containing 20lb of water is hanging...Ch. 6.6 - A rocket weighing 3 tons is filled with 40 tons of...Ch. 6.6 - It follows from Coulomb’s law in physics that...Ch. 6.6 - It follows from Newton’s Law of Universal...Ch. 6.6 - (a) The formula x=k/x2 in Exercise 25 is...Ch. 6.6 - The world’s first commercial high-speed magnetic...Ch. 6.6 - Assume that a Mars probe of mass m=2.00103kg is...Ch. 6.6 - On August 10,1972 a meteorite with an estimated...Ch. 6.7 - Prob. 1QCECh. 6.7 - A homogeneous lamina of mass M and density ...Ch. 6.7 - Let R be the region between the graphs of y=x2 and...Ch. 6.7 - If the region R in Quick Check Exercise 3 is used...Ch. 6.7 - Masses m1=5,m2=10, and m3=20 are positioned on a...Ch. 6.7 - Masses m1=10,m2=3,m3=4, and m are positioned on a...Ch. 6.7 - Find the centroid of the region by inspection and...Ch. 6.7 - Find the centroid of the region by inspection and...Ch. 6.7 - Find the centroid of the region by inspection and...Ch. 6.7 - Find the centroid of the region by inspection and...Ch. 6.7 - Find the centroid of the region.Ch. 6.7 - Find the centroid of the region.Ch. 6.7 - Find the centroid of the region.Ch. 6.7 - Find the centroid of the region.Ch. 6.7 - Find the centroid of the region. The triangle with...Ch. 6.7 - Find the centroid of the region. The region...Ch. 6.7 - Find the centroid of the region. The region...Ch. 6.7 - Find the centroid of the region. The region...Ch. 6.7 - Find the centroid of the region. The region...Ch. 6.7 - Find the centroid of the region. The region...Ch. 6.7 - Use symmetry considerations to argue that the...Ch. 6.7 - Use symmetry considerations to argue that the...Ch. 6.7 - Find the mass and center of gravity of the lamina...Ch. 6.7 - Find the mass and center of gravity of the lamina...Ch. 6.7 - Find the mass and center of gravity of the lamina...Ch. 6.7 - Use a CAS to find the mass and center of gravity...Ch. 6.7 - Use a CAS to find the mass and center of gravity...Ch. 6.7 - Use a CAS to find the mass and center of gravity...Ch. 6.7 - Determine whether the statement is true or false....Ch. 6.7 - Determine whether the statement is true or false....Ch. 6.7 - Determine whether the statement is true or false....Ch. 6.7 - Determine whether the statement is true or false....Ch. 6.7 - Find the centroid of the triangle with vertices...Ch. 6.7 - Prove that the centroid of a triangle is the point...Ch. 6.7 - Find the centroid of the isosceles trapezoid with...Ch. 6.7 - Prove that the centroid of a parallelogram is the...Ch. 6.7 - Use the Theorem of Pappus and the fact that the...Ch. 6.7 - Use the Theorem of Pappus and the fact that the...Ch. 6.7 - Use the Theorem of Pappus to find the volume of...Ch. 6.7 - Use the Theorem of Pappus to find the centroid of...Ch. 6.8 - The pressure unit equivalent to a newton per...Ch. 6.8 - Given that the weight density of water is 9810N/m3...Ch. 6.8 - Suppose that a flat surface is immersed vertically...Ch. 6.8 - A rectangular plate 2m wide and 3m high is...Ch. 6.8 - In this exercise set, refer to Table 6.8.2 for...Ch. 6.8 - (a) Find the force (in N ) on the deck of a sunken...Ch. 6.8 - The flat surfaces shown are submerged vertically...Ch. 6.8 - The flat surfaces shown are submerged vertically...Ch. 6.8 - The flat surfaces shown are submerged vertically...Ch. 6.8 - The flat surfaces shown are submerged vertically...Ch. 6.8 - The flat surfaces shown are submerged vertically...Ch. 6.8 - The flat surfaces shown are submerged vertically...Ch. 6.8 - Suppose that a flat surface is immersed vertically...Ch. 6.8 - An oil tank is shaped like a right circular...Ch. 6.8 - A square plate of side a feet is dipped in a...Ch. 6.8 - True-False Determine whether the statement is true...Ch. 6.8 - True-False Determine whether the statement is true...Ch. 6.8 - True-False Determine whether the statement is true...Ch. 6.8 - True-False Determine whether the statement is true...Ch. 6.8 - Formula 8 gives the fluid force on a flat surface...Ch. 6.8 - Prob. 17ESCh. 6.8 - Formula 8 gives the fluid force on a flat surface...Ch. 6.8 - Prob. 19ESCh. 6.8 - An observation window on a submarine is a square...Ch. 6.8 - (a) Show: If the submarine in Exercise 20 descends...Ch. 6.8 - (a) Let D=Da denote a disk of radius a submerged...Ch. 6.9 - coshx=sinhx=tanhx=Ch. 6.9 - Complete the table.Ch. 6.9 - Prob. 3QCECh. 6.9 - ddxcoshx=ddxsinhx=ddxtanhx=Ch. 6.9 - coshxdx=sinhxdx=tanhxdx=Ch. 6.9 - ddxcosh1x=ddxsinh1x=ddxtanh1x=Ch. 6.9 - Approximate the expression to four decimal places....Ch. 6.9 - Find the exact numerical value of each expression,...Ch. 6.9 - In each part, rewrite the expression as a ratio of...Ch. 6.9 - In each part, a value for one of the hyperbolic...Ch. 6.9 - Obtain the derivative formulas for cschx,sechx,...Ch. 6.9 - Find the derivatives of cosh1x and tanh1x , and...Ch. 6.9 - Find the derivatives of sinh1x,cosh1x,tanh1x by...Ch. 6.9 - Find dy/dx . y=sinh4x8Ch. 6.9 - Find dy/dx . y=coshx4Ch. 6.9 - Find dy/dx . y=cothlnxCh. 6.9 - Find dy/dx 12. y=ln(tanh5x)Ch. 6.9 - Find dy/dx . y=csch1/xCh. 6.9 - Find dy/dx . y=seche2xCh. 6.9 - Find dy/dx . y=4x+cosh25xCh. 6.9 - Find dy/dx 16. y=sinh4(3x)Ch. 6.9 - Find dy/dx . y=x3tanh2xCh. 6.9 - Find dy/dx . y=sinhcos3xCh. 6.9 - Find dy/dx . y=sinh113xCh. 6.9 - Find dy/dx . y=sinh11/xCh. 6.9 - Prob. 21ESCh. 6.9 - Find dy/dx . y=cosh1sinh1xCh. 6.9 - Find dy/dx . y=coth1x2Ch. 6.9 - Prob. 25ESCh. 6.9 - Prob. 26ESCh. 6.9 - Prob. 27ESCh. 6.9 - Prob. 28ESCh. 6.9 - Evaluate the integrals. sinh6xcoshxdxCh. 6.9 - Evaluate the integrals. 30. cosh(3x+2)dxCh. 6.9 - Evaluate the integrals. tanhxsech2xdxCh. 6.9 - Evaluate the integrals. csch23xdxCh. 6.9 - Evaluate the integrals. tanhxdxCh. 6.9 - Prob. 34ESCh. 6.9 - Evaluate the integrals. ln2ln3tanhxsech3xdxCh. 6.9 - Evaluate the integrals. 0ln3exexex+exdxCh. 6.9 - Evaluate the integrals. dx1+9x2Ch. 6.9 - Evaluate the integrals. dxx22x2Ch. 6.9 - Evaluate the integrals. dx1e2xx0Ch. 6.9 - Evaluate the integrals. sind1+cos2Ch. 6.9 - Evaluate the integrals. dxx1+4x2Ch. 6.9 - Prob. 42ESCh. 6.9 - Prob. 43ESCh. 6.9 - Evaluate the integrals. 03dtt2+1Ch. 6.9 - True-False Determine whether the statement is true...Ch. 6.9 - True-False Determine whether the statement is true...Ch. 6.9 - True-False Determine whether the statement is true...Ch. 6.9 - True-False Determine whether the statement is true...Ch. 6.9 - Find the area enclosed by y=sinh2x,y=0, and x=ln3...Ch. 6.9 - Find the volume of the solid that is generated...Ch. 6.9 - Find the arc length of the catenary y=coshx...Ch. 6.9 - Find the arc length of the catenary y=acoshx/a...Ch. 6.9 - In parts (a)-(f) find the limits, and confirm that...Ch. 6.9 - Explain how to obtain the asymptotes for y=tanhx...Ch. 6.9 - Prove that sinhx is an odd function of x and that...Ch. 6.9 - Prove the identities. (a) coshx+sinhx=ex (b)...Ch. 6.9 - Prove the identities. (a) 1tanh2x=sech2x (b)...Ch. 6.9 - Prove: (a) cosh1x=lnx+x21,x1 (b)...Ch. 6.9 - Use Exercise 60 to obtain the derivative formulas...Ch. 6.9 - Prove:...Ch. 6.9 - Use Exercise 62 to express the integral dy1u2...Ch. 6.9 - Show that (a) ddxsech1x=1x1+x2 (b)...Ch. 6.9 - In each part, find the limit. (a) limx+cosh1xlnx...Ch. 6.9 - Use the first and second derivatives to show that...Ch. 6.9 - The integration formulas for 1/u2a2 in Theorem...Ch. 6.9 - Show that sinhx+coshxn=sinhnx+coshnx .Ch. 6.9 - Show that aaetxdx=2sinattCh. 6.9 - A cable is suspended between two poles as shown in...Ch. 6.9 - These exercises refer to the hanging cable...Ch. 6.9 - These exercises refer to the hanging cable...Ch. 6.9 - The design of the Gateway Arch in St. Louis,...Ch. 6.9 - Suppose that a hollow tube rotates with a constant...Ch. 6.9 - The accompanying figure shows a person pulling a...Ch. 6.9 - The death rate for victims of a plague during the...Ch. 6 - Describe the method of slicing for finding...Ch. 6 - State an integral formula for finding a volume by...Ch. 6 - State an integral formula for finding the arc...Ch. 6 - State an integral formula for the work W done by a...Ch. 6 - State an integral formula for the fluid force F...Ch. 6 - Let R be the region in the first quadrant enclosed...Ch. 6 - (a) Set up a sum of definite integrals that...Ch. 6 - The accompanying figure shows velocity versus time...Ch. 6 - Let R be the region enclosed by the curves...Ch. 6 - A football has the shape of the solid generated by...Ch. 6 - Find the volume of the solid whose base is the...Ch. 6 - Find the arc length in the second quadrant of the...Ch. 6 - Let C be the curve y=ex between x=0 and x=ln10 ....Ch. 6 - Find the area of the surface generated by...Ch. 6 - Consider the solid generated by revolving the...Ch. 6 - (a) A spring exerts a force of 0.5N when stretched...Ch. 6 - A boat is anchored so that the anchor is 150ft...Ch. 6 - Find the centroid of the region. The region...Ch. 6 - Find the centroid of the region. The upper half of...Ch. 6 - In each part, set up, but do not evaluate, an...Ch. 6 - Show that for any constant a, the function...Ch. 6 - In each part, prove the identity. (a)...
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- Q2: Find the interval and radius of convergence for the following series: Σ n=1 (-1)η-1 xn narrow_forward8. Evaluate arctan x dx a) xartanx 2 2 In(1 + x²) + C b) xartanx + 1½-3ln(1 + x²) + C c) xartanx + In(1 + x²) + C d) (arctanx)² + C 2 9) Evaluate Inx³ dx 3 a) +C b) ln x² + C c)¾½ (lnx)² d) 3x(lnx − 1) + C - x 10) Determine which integral is obtained when the substitution x = So¹² √1 - x²dx sine is made in the integral πT π π a) √ sin cos e de b) √ cos² de c) c Ꮎ Ꮎ cos² 0 de c) cos e de d) for cos² e de πT 11. Evaluate tan³xdx 1 a) b) c) [1 - In 2] 2 2 c) [1 − In2] d)½½[1+ In 2]arrow_forward12. Evaluate ſ √9-x2 -dx. x2 a) C 9-x2 √9-x2 - x2 b) C - x x arcsin ½-½ c) C + √9 - x² + arcsin x d) C + √9-x2 x2 13. Find the indefinite integral S cos³30 √sin 30 dᎾ . 2√√sin 30 (5+sin²30) √sin 30 (3+sin²30) a) C+ √sin 30(5-sin²30) b) C + c) C + 5 5 5 10 d) C + 2√√sin 30 (3-sin²30) 2√√sin 30 (5-sin²30) e) C + 5 15 14. Find the indefinite integral ( sin³ 4xcos 44xdx. a) C+ (7-5cos24x)cos54x b) C (7-5cos24x)cos54x (7-5cos24x)cos54x - 140 c) C - 120 140 d) C+ (7-5cos24x)cos54x e) C (7-5cos24x)cos54x 4 4 15. Find the indefinite integral S 2x2 dx. ex - a) C+ (x²+2x+2)ex b) C (x² + 2x + 2)e-* d) C2(x²+2x+2)e¯* e) C + 2(x² + 2x + 2)e¯* - c) C2x(x²+2x+2)e¯*arrow_forward
- 4. Which substitution would you use to simplify the following integrand? S a) x = sin b) x = 2 tan 0 c) x = 2 sec 3√√3 3 x3 5. After making the substitution x = = tan 0, the definite integral 2 2 3 a) ៖ ស្លឺ sin s π - dᎾ 16 0 cos20 b) 2/4 10 cos 20 π sin30 6 - dᎾ c) Π 1 cos³0 3 · de 16 0 sin20 1 x²√x²+4 3 (4x²+9)2 π d) cos²8 16 0 sin³0 dx d) x = tan 0 dx simplifies to: de 6. In order to evaluate (tan 5xsec7xdx, which would be the most appropriate strategy? a) Separate a sec²x factor b) Separate a tan²x factor c) Separate a tan xsecx factor 7. Evaluate 3x x+4 - dx 1 a) 3x+41nx + 4 + C b) 31n|x + 4 + C c) 3 ln x + 4+ C d) 3x - 12 In|x + 4| + C x+4arrow_forward1. Abel's Theorem. The goal in this problem is to prove Abel's theorem by following a series of steps (each step must be justified). Theorem 0.1 (Abel's Theorem). If y1 and y2 are solutions of the differential equation y" + p(t) y′ + q(t) y = 0, where p and q are continuous on an open interval, then the Wronskian is given by W (¥1, v2)(t) = c exp(− [p(t) dt), where C is a constant that does not depend on t. Moreover, either W (y1, y2)(t) = 0 for every t in I or W (y1, y2)(t) = 0 for every t in I. 1. (a) From the two equations (which follow from the hypotheses), show that y" + p(t) y₁ + q(t) y₁ = 0 and y½ + p(t) y2 + q(t) y2 = 0, 2. (b) Observe that Hence, conclude that (YY2 - Y1 y2) + P(t) (y₁ Y2 - Y1 Y2) = 0. W'(y1, y2)(t) = yY2 - Y1 y2- W' + p(t) W = 0. 3. (c) Use the result from the previous step to complete the proof of the theorem.arrow_forward2. Observations on the Wronskian. Suppose the functions y₁ and y2 are solutions to the differential equation p(x)y" + q(x)y' + r(x) y = 0 on an open interval I. 1. (a) Prove that if y₁ and y2 both vanish at the same point in I, then y₁ and y2 cannot form a fundamental set of solutions. 2. (b) Prove that if y₁ and y2 both attain a maximum or minimum at the same point in I, then y₁ and Y2 cannot form a fundamental set of solutions. 3. (c) show that the functions & and t² are linearly independent on the interval (−1, 1). Verify that both are solutions to the differential equation t² y″ – 2ty' + 2y = 0. Then justify why this does not contradict Abel's theorem. 4. (d) What can you conclude about the possibility that t and t² are solutions to the differential equation y" + q(x) y′ + r(x)y = 0?arrow_forward
- Question 4 Find an equation of (a) The plane through the point (2, 0, 1) and perpendicular to the line x = y=2-t, z=3+4t. 3t, (b) The plane through the point (3, −2, 8) and parallel to the plane z = x+y. (c) The plane that contains the line x = 1+t, y = 2 − t, z = 4 - 3t and is parallel to the plane 5x + 2y + z = 1. (d) The plane that passes through the point (1,2,3) and contains the line x = 3t, y = 1+t, and z = 2-t. (e) The plane that contains the lines L₁: x = 1 + t, y = 1 − t, z = 2t and L2 : x = 2 − s, y = s, z = 2.arrow_forwardPlease find all values of x.arrow_forward3. Consider the initial value problem 9y" +12y' + 4y = 0, y(0) = a>0: y′(0) = −1. Solve the problem and find the value of a such that the solution of the initial value problem is always positive.arrow_forward
- 5. Euler's equation. Determine the values of a for which all solutions of the equation 5 x²y" + axy' + y = 0 that have the form (A + B log x) x* or Ax¹¹ + Bä” tend to zero as a approaches 0.arrow_forward4. Problem on variable change. The purpose of this problem is to perform an appropriate change of variables in order to reduce the problem to a second-order equation with constant coefficients. ty" + (t² − 1)y'′ + t³y = 0, 0arrow_forward4. Some psychologists contend that the number of facts of a certain type that are remembered after t hours is given by f(t)== 90t 951-90 Find the rate at which the number of facts remembered is changing after 1 hour and after 10 hours. Interpret.arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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