Calculus (MindTap Course List)
8th Edition
ISBN: 9781285740621
Author: James Stewart
Publisher: Cengage Learning
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Question
Chapter 5.P, Problem 7P
To determine
To show:
The depth of the water decreases at a constant rate regardless of the shape of the bowl.
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Water in an open bowl evaporates at a rate proportional to the area of the surface of the water. (This means that the rate of decrease of the volume is proportional to the area of the surface.) Show that the depth of the water decreases at a constant rate, regardless of the shape of the bowl.
Chapter 5 Solutions
Calculus (MindTap Course List)
Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Find the area of the shaded region.Ch. 5.1 - Prob. 3ECh. 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 - Prob. 10E
Ch. 5.1 - Sketch the region enclosed by the given curves....Ch. 5.1 - Prob. 12ECh. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Prob. 14ECh. 5.1 - Prob. 15ECh. 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. 19ECh. 5.1 - Sketch the region enclosed by the given curves and...Ch. 5.1 - Prob. 21ECh. 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 - Prob. 26ECh. 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. 29ECh. 5.1 - Sketch the region enclosed by the given curves and...Ch. 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 - Use a graph to find approximate x-coordinates of...Ch. 5.1 - Prob. 38ECh. 5.1 - Prob. 39ECh. 5.1 - Prob. 40ECh. 5.1 - Prob. 41ECh. 5.1 - Prob. 42ECh. 5.1 - Graph the region between the curves and use your...Ch. 5.1 - Prob. 44ECh. 5.1 - Prob. 45ECh. 5.1 - Prob. 46ECh. 5.1 - Prob. 47ECh. 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 - Prob. 51ECh. 5.1 - Prob. 52ECh. 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 - 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 - Prob. 60ECh. 5.1 - Sketch the region bounded by the given curves and...Ch. 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 - Prob. 3ECh. 5.2 - Prob. 4ECh. 5.2 - Find the volume of the solid obtained by rotating...Ch. 5.2 - Prob. 6ECh. 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 - 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. 16ECh. 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 - 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 - Prob. 26ECh. 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 - Prob. 31ECh. 5.2 - Prob. 32ECh. 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. 35ECh. 5.2 - Prob. 36ECh. 5.2 - Prob. 37ECh. 5.2 - Prob. 38ECh. 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 - 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 - Prob. 55ECh. 5.2 - Prob. 56ECh. 5.2 - Find the volume of the described solid S. The base...Ch. 5.2 - Prob. 58ECh. 5.2 - Prob. 59ECh. 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 - Prob. 62ECh. 5.2 - a Set up an integral for the volume of a solid...Ch. 5.2 - Prob. 64ECh. 5.2 - Prob. 65ECh. 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 - Prob. 69ECh. 5.2 - Prob. 70ECh. 5.2 - Prob. 71ECh. 5.2 - Prob. 72ECh. 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 - Prob. 9ECh. 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 - Prob. 14ECh. 5.3 - Prob. 15ECh. 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 - Prob. 20ECh. 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 - a Set up an integral for the volume of the solid...Ch. 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 - Use a computer algebra system to find the exact...Ch. 5.3 - Prob. 36ECh. 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 - Prob. 39ECh. 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 - Prob. 45ECh. 5.3 - Prob. 46ECh. 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 - Prob. 2ECh. 5.4 - Prob. 3ECh. 5.4 - Prob. 4ECh. 5.4 - Prob. 5ECh. 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 - Prob. 12ECh. 5.4 - Prob. 13ECh. 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 - Prob. 17ECh. 5.4 - Show how to approximate the required work by a...Ch. 5.4 - Prob. 19ECh. 5.4 - Prob. 20ECh. 5.4 - Show how to approximate the required work by a...Ch. 5.4 - Prob. 22ECh. 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 - Prob. 29ECh. 5.4 - In a steam engine the pressure P and volume V of...Ch. 5.4 - Prob. 31ECh. 5.4 - Prob. 32ECh. 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 - Prob. 3ECh. 5.5 - Prob. 4ECh. 5.5 - Find the average value of the function on the...Ch. 5.5 - Prob. 6ECh. 5.5 - Find the average value of the function on the...Ch. 5.5 - Prob. 8ECh. 5.5 - a Find the average value of f on the given...Ch. 5.5 - Prob. 10ECh. 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 - Prob. 15ECh. 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 - Prob. 19ECh. 5.5 - If a freely falling body starts from rest, then...Ch. 5.5 - Prob. 21ECh. 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 - Prob. 24ECh. 5.R - a Draw two typical curves y=f(x) and y=g(x) where...Ch. 5.R - Prob. 2CCCh. 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 - Prob. 6CCCh. 5.R - Prob. 1ECh. 5.R - Prob. 2ECh. 5.R - Prob. 3ECh. 5.R - Prob. 4ECh. 5.R - Prob. 5ECh. 5.R - Prob. 6ECh. 5.R - Prob. 7ECh. 5.R - Find the volume of the solid obtained by rotating...Ch. 5.R - Prob. 9ECh. 5.R - Prob. 10ECh. 5.R - Prob. 11ECh. 5.R - Set up, but do not evaluate, an integral for the...Ch. 5.R - Prob. 13ECh. 5.R - Prob. 14ECh. 5.R - Prob. 15ECh. 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 - Prob. 19ECh. 5.R - Prob. 20ECh. 5.R - Prob. 21ECh. 5.R - Prob. 22ECh. 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 - Prob. 27ECh. 5.R - Prob. 28ECh. 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 - Prob. 31ECh. 5.R - Prob. 32ECh. 5.R - Prob. 33ECh. 5.R - Prob. 34ECh. 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 - Prob. 7PCh. 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 - Prob. 15P
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