Student Solutions Manual, Chapters 1-11 for Stewart's Single Variable Calculus, 8th (James Stewart Calculus)
8th Edition
ISBN: 9781305271814
Author: James Stewart
Publisher: Cengage Learning
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Textbook Question
Chapter 5, Problem 12P
A cylindrical container of radius r and height L is partially filled with a liquid whose volume is V. If the container is rotated about its axis of symmetry with constant angular speed ω, then the container will induce a rotational motion in the liquid around the same axis. Eventually, the liquid will be rotating at the same angular speed as the container. The surface of the liquid will be convex, as indicated in the figure, because the centrifugal force on the liquid particles increases with the distance from the axis of the container. It can be shown that the surface of the liquid is a paraboloid of revolution generated by rotating the parabola
about the y-axis, where g is the acceleration due to gravity.
(a) Determine h as a function of ω.
(b) At what angular speed will the surface of the liquid touch the bottom? At what speed will it spill over the top?
(c) Suppose the radius of the container is 2 ft, the height is 7 ft, and the container and liquid are rotating at the same constant angular speed. The surface of the liquid is 5 ft below the top of the tank at the central axis and 4 ft below the top of the tank 1 ft out from the central axis.
(i) Determine the angular speed of the container and the volume of the fluid.
(ii) How far below the top of the tank is the liquid at the wall of the container?
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Chapter 5 Solutions
Student Solutions Manual, Chapters 1-11 for Stewart's Single Variable Calculus, 8th (James Stewart Calculus)
Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Prob. 6ECh. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Prob. 8ECh. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves....
Ch. 5.1 - Prob. 11ECh. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Prob. 13ECh. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Prob. 17ECh. 5.1 - Prob. 18ECh. 5.1 - Prob. 19ECh. 5.1 - Prob. 20ECh. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Prob. 22ECh. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Prob. 25ECh. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Prob. 27ECh. 5.1 - Prob. 28ECh. 5.1 - Prob. 29ECh. 5.1 - Prob. 30ECh. 5.1 - Prob. 31ECh. 5.1 - Prob. 32ECh. 5.1 - Prob. 33ECh. 5.1 - Prob. 34ECh. 5.1 - Prob. 35ECh. 5.1 - Evaluate the integral and interpret it as the area...Ch. 5.1 - Prob. 37ECh. 5.1 - Prob. 38ECh. 5.1 - Prob. 39ECh. 5.1 - Prob. 40ECh. 5.1 - Prob. 41ECh. 5.1 - Graph the region between the curves and use your...Ch. 5.1 - Prob. 43ECh. 5.1 - Prob. 44ECh. 5.1 - Use a computer algebra system to find the exact...Ch. 5.1 - Prob. 46ECh. 5.1 - Racing cars driven by Chris and Kelly are side by...Ch. 5.1 - The widths (in meters) of a kidney-shaped swimming...Ch. 5.1 - A cross-section of an airplane wing is shown....Ch. 5.1 - If the birth rate of a population is b(t) = 2200 +...Ch. 5.1 - In Example 5, we modeled a measles pathogenesis...Ch. 5.1 - The rates at which rain fell, in inches per hour,...Ch. 5.1 - Two cars, A and B, start side by side and...Ch. 5.1 - The figure shows graphs of the marginal revenue...Ch. 5.1 - Prob. 55ECh. 5.1 - Prob. 56ECh. 5.1 - Prob. 57ECh. 5.1 - (a) Find the number a such that the line x = a...Ch. 5.1 - Find the values of c such that the area of the...Ch. 5.1 - Prob. 60ECh. 5.1 - Prob. 61ECh. 5.1 - Prob. 62ECh. 5.1 - Prob. 63ECh. 5.1 - For what values of m do the line y = mx and the...Ch. 5.2 - Find the volume of the solid obtained by rotating...Ch. 5.2 - Find the volume of the solid obtained by rotating...Ch. 5.2 - Find the volume of the solid obtained by rotating...Ch. 5.2 - Find the volume of the solid obtained by rotating...Ch. 5.2 - Find the volume of the solid obtained by rotating...Ch. 5.2 - Find the volume of the solid obtained by rotating...Ch. 5.2 - Find the volume of the solid obtained by rotating...Ch. 5.2 - Prob. 8ECh. 5.2 - Prob. 9ECh. 5.2 - Prob. 10ECh. 5.2 - Prob. 11ECh. 5.2 - Prob. 12ECh. 5.2 - Find the volume of the solid obtained by rotating...Ch. 5.2 - Find the volume of the solid obtained by rotating...Ch. 5.2 - Prob. 15ECh. 5.2 - Find the volume of the solid obtained by rotating...Ch. 5.2 - Find the volume of the solid obtained by rotating...Ch. 5.2 - Prob. 18ECh. 5.2 - Refer to the figure and find the volume generated...Ch. 5.2 - Refer to the figure and find the volume generated...Ch. 5.2 - Refer to the figure and find the volume generated...Ch. 5.2 - Prob. 22ECh. 5.2 - Refer to the figure and find the volume generated...Ch. 5.2 - Prob. 24ECh. 5.2 - Refer to the figure and find the volume generated...Ch. 5.2 - Refer to the figure and find the volume generated...Ch. 5.2 - Refer to the figure and find the volume generated...Ch. 5.2 - Prob. 28ECh. 5.2 - Refer to the figure and find the volume generated...Ch. 5.2 - Prob. 30ECh. 5.2 - Set up an integral for the volume of the solid...Ch. 5.2 - Set up an integral for the volume of the solid...Ch. 5.2 - Set up an integral for the volume of the solid...Ch. 5.2 - Prob. 34ECh. 5.2 - Prob. 35ECh. 5.2 - Prob. 36ECh. 5.2 - Use a computer algebra system to find the exact...Ch. 5.2 - Use a computer algebra system to find the exact...Ch. 5.2 - Prob. 39ECh. 5.2 - Prob. 40ECh. 5.2 - Prob. 41ECh. 5.2 - Prob. 42ECh. 5.2 - A CAT scan produces equally spaced cross-sectional...Ch. 5.2 - Prob. 44ECh. 5.2 - Prob. 45ECh. 5.2 - Prob. 46ECh. 5.2 - Prob. 47ECh. 5.2 - Find the volume of the described solid S. 48.A...Ch. 5.2 - Find the volume of the described solid S. 49.A cap...Ch. 5.2 - Find the volume of the described solid S. 50. A...Ch. 5.2 - Prob. 51ECh. 5.2 - Find the volume of the described solid S. 52. A...Ch. 5.2 - Prob. 53ECh. 5.2 - Find the volume of the described solid S. 54. The...Ch. 5.2 - Find the volume of the described solid S. 55. The...Ch. 5.2 - Prob. 56ECh. 5.2 - Prob. 57ECh. 5.2 - Prob. 58ECh. 5.2 - Prob. 59ECh. 5.2 - Prob. 60ECh. 5.2 - Prob. 61ECh. 5.2 - The base of S is a circular disk with radius r....Ch. 5.2 - (a) Set up an integral for the volume of a solid...Ch. 5.2 - Prob. 64ECh. 5.2 - (a) Cavalieris Principle states that if a family...Ch. 5.2 - Find the volume common to two circular cylinders,...Ch. 5.2 - Prob. 67ECh. 5.2 - A bowl is shaped like a hemisphere with diameter...Ch. 5.2 - A hole of radius r is bored through the middle of...Ch. 5.2 - A hole of radius r is bored through the center of...Ch. 5.2 - Prob. 71ECh. 5.2 - Suppose that a region R has area A and lies above...Ch. 5.3 - Let S be the solid obtained by rotating the region...Ch. 5.3 - Let S be the solid obtained by rotating the region...Ch. 5.3 - Prob. 3ECh. 5.3 - Prob. 4ECh. 5.3 - Prob. 5ECh. 5.3 - Prob. 6ECh. 5.3 - Prob. 7ECh. 5.3 - Let V be the volume of the solid obtained by...Ch. 5.3 - Use the method of cylindrical shells to find the...Ch. 5.3 - Use the method of cylindrical shells to find the...Ch. 5.3 - Use the method of cylindrical shells to find the...Ch. 5.3 - Use the method of cylindrical shells to find the...Ch. 5.3 - Use the method of cylindrical shells to find the...Ch. 5.3 - Use the method of cylindrical shells to find the...Ch. 5.3 - Use the method of cylindrical shells to find the...Ch. 5.3 - Use the method of cylindrical shells to find the...Ch. 5.3 - Use the method of cylindrical shells to find the...Ch. 5.3 - Use the method of cylindrical shells to find the...Ch. 5.3 - Use the method of cylindrical shells to find the...Ch. 5.3 - Use the method of cylindrical shells to find the...Ch. 5.3 - (a) Set up an integral for the volume of the solid...Ch. 5.3 - Prob. 22ECh. 5.3 - Prob. 23ECh. 5.3 - Prob. 24ECh. 5.3 - Prob. 25ECh. 5.3 - Prob. 26ECh. 5.3 - Prob. 27ECh. 5.3 - If the region shown in the figure is rotated about...Ch. 5.3 - Prob. 29ECh. 5.3 - Prob. 30ECh. 5.3 - Prob. 31ECh. 5.3 - Prob. 32ECh. 5.3 - Prob. 33ECh. 5.3 - Prob. 34ECh. 5.3 - Prob. 35ECh. 5.3 - Use a computer algebra system to find the exact...Ch. 5.3 - Prob. 37ECh. 5.3 - Prob. 38ECh. 5.3 - Prob. 39ECh. 5.3 - Prob. 40ECh. 5.3 - Prob. 41ECh. 5.3 - Prob. 42ECh. 5.3 - Prob. 43ECh. 5.3 - Let T be the triangular region with vertices (0,...Ch. 5.3 - Prob. 45ECh. 5.3 - Use cylindrical shells to find the volume of the...Ch. 5.3 - Prob. 47ECh. 5.3 - Suppose you make napkin rings by drilling holes...Ch. 5.4 - A 360-lb gorilla climbs a tree to a height of 20...Ch. 5.4 - How much work is done when a hoist lifts a 200-kg...Ch. 5.4 - Prob. 3ECh. 5.4 - Prob. 4ECh. 5.4 - Shown is the graph of a force function (in...Ch. 5.4 - Prob. 6ECh. 5.4 - Prob. 7ECh. 5.4 - A spring has a natural length of 40 cm. If a 60-N...Ch. 5.4 - Suppose that 2 J of work is needed to stretch a...Ch. 5.4 - Prob. 10ECh. 5.4 - Prob. 11ECh. 5.4 - Prob. 12ECh. 5.4 - Prob. 13ECh. 5.4 - Prob. 14ECh. 5.4 - Prob. 15ECh. 5.4 - Prob. 16ECh. 5.4 - Show how to approximate the required work by a...Ch. 5.4 - Prob. 18ECh. 5.4 - Prob. 19ECh. 5.4 - Show how to approximate the required work by a...Ch. 5.4 - Prob. 21ECh. 5.4 - Show how to approximate the required work by a...Ch. 5.4 - A tank is full of water. Find the work required to...Ch. 5.4 - A tank is full of water. Find the work required to...Ch. 5.4 - A tank is full of water. Find the work required to...Ch. 5.4 - A tank is full of water. Find the work required to...Ch. 5.4 - Prob. 27ECh. 5.4 - Prob. 28ECh. 5.4 - When gas expands in a cylinder with radius r, the...Ch. 5.4 - In a steam engine the pressure P and volume V of...Ch. 5.4 - The kinetic energy KE of an object of mass m...Ch. 5.4 - Suppose that when launching an 800-kg roller...Ch. 5.4 - Prob. 33ECh. 5.4 - The Great Pyramid of King Khufu was built of...Ch. 5.5 - Find the average value of the function on the...Ch. 5.5 - Prob. 2ECh. 5.5 - Find the average value of the function on the...Ch. 5.5 - Prob. 4ECh. 5.5 - Prob. 5ECh. 5.5 - Prob. 6ECh. 5.5 - Prob. 7ECh. 5.5 - Prob. 8ECh. 5.5 - (a) Find the average value of f on the given...Ch. 5.5 - Prob. 10ECh. 5.5 - Prob. 11ECh. 5.5 - Prob. 12ECh. 5.5 - If f is continuous and 13f(x)dx=8, show that f...Ch. 5.5 - Prob. 14ECh. 5.5 - Find the average value of f on [0, 8].Ch. 5.5 - The velocity graph of an accelerating car is...Ch. 5.5 - Prob. 17ECh. 5.5 - The velocity v of blood that flows in a blood...Ch. 5.5 - The linear density in a rod 8 m long is...Ch. 5.5 - Prob. 20ECh. 5.5 - Prob. 21ECh. 5.5 - Use the diagram to show that if f is concave...Ch. 5.5 - Prob. 23ECh. 5.5 - Prob. 24ECh. 5 - (a) Draw two typical curves y = f(x) and y = g(x),...Ch. 5 - Prob. 2RCCCh. 5 - Prob. 3RCCCh. 5 - Prob. 4RCCCh. 5 - Prob. 5RCCCh. 5 - Prob. 6RCCCh. 5 - Prob. 1RECh. 5 - Find the area of the region bounded by the given...Ch. 5 - Find the area of the region bounded by the given...Ch. 5 - Prob. 4RECh. 5 - Prob. 5RECh. 5 - Prob. 6RECh. 5 - Prob. 7RECh. 5 - Prob. 8RECh. 5 - Prob. 9RECh. 5 - Prob. 10RECh. 5 - Find the volume of the solid obtained by rotating...Ch. 5 - Set up, but do not evaluate, an integral for the...Ch. 5 - Prob. 13RECh. 5 - Set up, but do not evaluate, an integral for the...Ch. 5 - Find the volumes of the solids obtained by...Ch. 5 - Let R be the region in the first quadrant bounded...Ch. 5 - Prob. 17RECh. 5 - Let R be the region bounded by the curves y = 1 ...Ch. 5 - Prob. 19RECh. 5 - Prob. 20RECh. 5 - Prob. 21RECh. 5 - Prob. 22RECh. 5 - Prob. 23RECh. 5 - Prob. 24RECh. 5 - The height of a monument is 20 m. A horizontal...Ch. 5 - Prob. 26RECh. 5 - Prob. 27RECh. 5 - Prob. 28RECh. 5 - A tank full of water has the shape of a paraboloid...Ch. 5 - A steel tank has the shape of a circular cylinder...Ch. 5 - Prob. 31RECh. 5 - (a) Find the average value of the function...Ch. 5 - If f is a continuous function, what is the limit...Ch. 5 - Let R1 be the region bounded by y = x2, y = 0, and...Ch. 5 - Prob. 1PCh. 5 - There is a line through the origin that divides...Ch. 5 - The figure shows a horizontal line y = c...Ch. 5 - A cylindrical glass of radius r and height L is...Ch. 5 - (a) Show that the volume of a segment of height h...Ch. 5 - Prob. 6PCh. 5 - Prob. 7PCh. 5 - Prob. 8PCh. 5 - The figure shows a curve C with the property that,...Ch. 5 - A paper drinking cup filled with water has the...Ch. 5 - A clepsydra, or water clock, is a glass container...Ch. 5 - A cylindrical container of radius r and height L...Ch. 5 - Prob. 13PCh. 5 - If the tangent at a point P on the curve y = x3...
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