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8th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781285740621

Chapter 5.R, Problem 3CC

Textbook Problem

(a) Suppose *S* is a solid with known cross-sectional areas. Explain how to approximate the volume of *S* by a Riemann sum. Then write an expression for the exact volume.

(b) If *S* is a solid of revolution, how do you find the cross-sectional areas?

Calculus (MindTap Course List)

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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 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves....

Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Sketch the region enclosed by the given curves....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 - 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 - 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 - 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 - 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 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - The graphs of two functions are shown with the...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 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Use calculus to find the area of the triangle with...Ch. 5.1 - Use calculus to find the area of the triangle with...Ch. 5.1 - Evaluate the integral and interpret it as the area...Ch. 5.1 - Evaluate the integral and interpret it as the area...Ch. 5.1 - Use a graph to find approximate x-coordinates of...Ch. 5.1 - Use a graph to find approximate x-coordinates of...Ch. 5.1 - Use a graph to find approximate x-coordinates of...Ch. 5.1 - Use a graph to find approximate x-coordinates of...Ch. 5.1 - Graph the region between the curves and use your...Ch. 5.1 - Graph the region between the curves and use your...Ch. 5.1 - Graph the region between the curves and use your...Ch. 5.1 - Graph the region between the curves and use your...Ch. 5.1 - Use a computer algebra system to find the exact...Ch. 5.1 - Sketch the region in the xy-plane defined by the...Ch. 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...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 - The curve with equation y2=x2(x+3) is called...Ch. 5.1 - Find the area of the region bounded by the...Ch. 5.1 - Find the number b such that the line y=b divides...Ch. 5.1 - a Find the number a such that the line x=a bisects...Ch. 5.1 - Find the values of c such that the area of the...Ch. 5.1 - Suppose that 0c/2. For what value of c is the area...Ch. 5.1 - Sketch the region bounded by the given curves and...Ch. 5.1 - Sketch the region bounded by the given curves and...Ch. 5.1 - Sketch the region bounded by the given curves and...Ch. 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 - 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 - 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 - 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 - 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 - 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 - 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 - 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 - Set up an integral for the volume of the solid...Ch. 5.2 - Use a graph to find approximate x-coordinates of...Ch. 5.2 - Use a graph to find approximate x-coordinates of...Ch. 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 - Each integral represents the volume of a solid....Ch. 5.2 - Each integral represents the volume of a solid....Ch. 5.2 - Each integral represents the volume of a solid....Ch. 5.2 - Each integral represents the volume of a solid....Ch. 5.2 - A CAT scan produces equally spaced cross-sectional...Ch. 5.2 - A log 10 m long is cut at 1-meter intervals and...Ch. 5.2 - a If the region shown in the figure is rotated...Ch. 5.2 - a A model for the shape of a birds egg is obtained...Ch. 5.2 - Find the volume of the described solid S. A right...Ch. 5.2 - Find the volume of the described solid S. A...Ch. 5.2 - Find the volume of the described solid S. A cap of...Ch. 5.2 - Find the volume of the described solid S. A...Ch. 5.2 - Find the volume of the described solid S. A...Ch. 5.2 - Find the volume of the described solid S. A...Ch. 5.2 - Find the volume of described solid S. A...Ch. 5.2 - Find the volume of described solid S. The base of...Ch. 5.2 - Find the volume of the described solid S. The base...Ch. 5.2 - Find the volume of the described solid S. The base...Ch. 5.2 - Find the volume of the described solid S. The base...Ch. 5.2 - Find the volume of the described solid S. The base...Ch. 5.2 - Find the volume of the described solid S. The base...Ch. 5.2 - Find the volume of the described solid S. The base...Ch. 5.2 - Find the volume of the described solid S. The...Ch. 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 - Solve Example 9 taking cross-sections to be...Ch. 5.2 - a Cavalieris Principle states that if a family of...Ch. 5.2 - Find the volume common to two circular cylinders,...Ch. 5.2 - Find the volume common to two spheres, each with...Ch. 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 - Some of the pioneers of calculus, such as Kepler...Ch. 5.2 - Suppose that a region 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 - 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 - 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 - a Set up an integral for the volume of the solid...Ch. 5.3 - a Set up an integral for the volume of the solid...Ch. 5.3 - a Set up an integral for the volume of the solid...Ch. 5.3 - a Set up an integral for the volume of the solid...Ch. 5.3 - a Set up an integral for the volume of the solid...Ch. 5.3 - Use the Midpoint Rule with n=5 to estimate the...Ch. 5.3 - If the region shown in the figure is rotated about...Ch. 5.3 - Each integral represents the volume of a solid....Ch. 5.3 - Each integral represents the volume of a solid....Ch. 5.3 - Each integral represents the volume of a solid....Ch. 5.3 - Each integral represents the volume of a solid....Ch. 5.3 - Use a graph to estimate the x-coordinates of the...Ch. 5.3 - Use a graph to estimate the x-coordinates of the...Ch. 5.3 - Use a computer algebra system to find the exact...Ch. 5.3 - Use a computer algebra system to find the exact...Ch. 5.3 - The region bounded by the given curves is rotated...Ch. 5.3 - The region bounded by the given curves is rotated...Ch. 5.3 - The region bounded by the given curves is rotated...Ch. 5.3 - The region bounded by the given curves is rotated...Ch. 5.3 - The region bounded by the given curves is rotated...Ch. 5.3 - The region bounded by the given curves is rotated...Ch. 5.3 - The region bounded by the given curves is rotated...Ch. 5.3 - Let T be the triangular region with vertices...Ch. 5.3 - Use cylindrical shells to find the volume of the...Ch. 5.3 - Use cylindrical shells to find the volume of the...Ch. 5.3 - Use cylindrical shells to find the volume of the...Ch. 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 - A variable force of 5x2 pounds moves an object...Ch. 5.4 - When a particle is located a distance x meters...Ch. 5.4 - Shown is the graph of a force function in newtons...Ch. 5.4 - The table shows values of a force function f(x)...Ch. 5.4 - A force of 10 lb is required to hold a spring...Ch. 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 - If the work required to stretch a spring 1 ft...Ch. 5.4 - A spring has natural length 20 cm. Compare the...Ch. 5.4 - If 6 J of work is needed to stretch a spring from...Ch. 5.4 - Show how to approximate the required work by a...Ch. 5.4 - Show how to approximate the required work by a...Ch. 5.4 - Show how to approximate the required work by a...Ch. 5.4 - Show how to approximate the required work by a...Ch. 5.4 - Show how to approximate the required work by a...Ch. 5.4 - Show how to approximate the required work by a...Ch. 5.4 - Show how to approximate the required work by a...Ch. 5.4 - Show how to approximate the required work by a...Ch. 5.4 - Show how to approximate the required work by a...Ch. 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 - Suppose that for the tank in Exercise 23 the pump...Ch. 5.4 - Solve Exercise 24 if the tank is half full of oil...Ch. 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 - a Newtons Law of Gravitation states that two...Ch. 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 - Find the average value of the function on the...Ch. 5.5 - Find the average value of the function on the...Ch. 5.5 - Find the average value of the function on the...Ch. 5.5 - Find the average value of the function on the...Ch. 5.5 - Find the average value of the function on the...Ch. 5.5 - Find the average value of the function on the...Ch. 5.5 - Find the average value of the function on the...Ch. 5.5 - a Find the average value of f on the given...Ch. 5.5 - a Find the average value of f on the given...Ch. 5.5 - a Find the average value of f on the given...Ch. 5.5 - a Find the average value of f on the given...Ch. 5.5 - If f is continuous and 13f(x)dx=8, show that f...Ch. 5.5 - Find the numbers b such that the average value of...Ch. 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 - In a certain city the temperature (inF)t hours...Ch. 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 - If a freely falling body starts from rest, then...Ch. 5.5 - Use the result of Exercise 4.5.57 to compute the...Ch. 5.5 - Use the diagram to show that if f is concave...Ch. 5.5 - Prove that Mean value Theorem for Integrals by...Ch. 5.5 - If fave[a,b] denotes the average value of f on the...Ch. 5.R - a Draw two typical curves y=f(x) and y=g(x) where...Ch. 5.R - Suppose that Sue runs faster than Kathy throughout...Ch. 5.R - a Suppose S is a solid with known cross-sectional...Ch. 5.R - a What is the volume of a cylindrical shell? b...Ch. 5.R - Suppose that you push a book across a 6-meter-long...Ch. 5.R - a What is the average value of a function f on an...Ch. 5.R - Find the area of the region bounded by the given...Ch. 5.R - Find the area of the region bounded by the given...Ch. 5.R - Find the area of the region bounded by the given...Ch. 5.R - Find the area of the region bounded by the given...Ch. 5.R - Find the area of the region bounded by the given...Ch. 5.R - Find the area of the region bounded by the given...Ch. 5.R - Find the volume of the solid obtained by rotating...Ch. 5.R - Find the volume of the solid obtained by rotating...Ch. 5.R - Find the volume of the solid obtained by rotating...Ch. 5.R - Find the volume of the solid obtained by rotating...Ch. 5.R - Find the volume of the solid obtained by rotating...Ch. 5.R - Set up, but do not evaluate, an integral for the...Ch. 5.R - Set up, but do not evaluate, an integral for the...Ch. 5.R - Set up, but do not evaluate, an integral for the...Ch. 5.R - Find the volumes of the solids obtained by...Ch. 5.R - Let be the region in the first quadrant bounded...Ch. 5.R - Let be the region bounded by the curves...Ch. 5.R - Let be the region bounded by the curves y=1x2 and...Ch. 5.R - Each integral represents the volume of a solid....Ch. 5.R - Each integral represents the volume of a solid....Ch. 5.R - Each integral represents the volume of a solid....Ch. 5.R - Each integral represents the volume of a solid....Ch. 5.R - The base of a solid is a circular disk with radius...Ch. 5.R - The base of a solid is the region bounded by the...Ch. 5.R - The height of a monument is 20 m. A horizontal...Ch. 5.R - a The base of a solid is a square with vertices...Ch. 5.R - A force of 30 N is required to maintain a spring...Ch. 5.R - A 1600-lb elevator is suspended by a 200-ft cable...Ch. 5.R - A tank full of water has the shape of a paraboloid...Ch. 5.R - A steel tank has the shape of a circular cylinder...Ch. 5.R - Find the average value of the function f(t)=sec2t...Ch. 5.R - a Find the average value of the function f(x)=1/x...Ch. 5.R - If f is a continuous function, what is the limit...Ch. 5.R - Let 1 be the region bounded by y=x2,y=0, and x=b,...Ch. 5.P - a Find a positive continuous function f such that...Ch. 5.P - There is a line through the origin that divides...Ch. 5.P - The figure shows a horizontal line y=c...Ch. 5.P - A cylindrical glass of radius r and height L is...Ch. 5.P - a Show that the volume of a segment of height h of...Ch. 5.P - Archimedes Principle states that the buoyant force...Ch. 5.P - Water in an open bowl evaporates at a rate...Ch. 5.P - A sphere of radius 1 overlaps a smaller sphere of...Ch. 5.P - The figure shows a curve C with the property that,...Ch. 5.P - A paper drinking cup filled with water has the...Ch. 5.P - A clepsydra, or water clock, is a glass container...Ch. 5.P - A cylindrical container of radius r and height L...Ch. 5.P - Suppose the graph of a cubic polynomial intersects...Ch. 5.P - Suppose we are planning to make a taco from a...Ch. 5.P - If the tangent at a point P on the curve y=x3...

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