Single Variable Essential Calculus: Early Transcendentals
Single Variable Essential Calculus: Early Transcendentals
2nd Edition
ISBN: 9781133112785
Author: James Stewart
Publisher: Cengage Learning
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Chapter 7, Problem 48RE

Let ℛ1, be the region bounded by y = x2, y = 0, and x = b, where b > 0. Let ℛ 2, be the region bounded by y = x2, x = 0, and y = b2.

(a) Is there a value of b such that ℛ 1, and ℛ 2, have the same area?

(b) Is there a value of b such that ℛ 1, sweeps out the same volume when rotated about the x-axis and the y-axis?

(c) Is there a value of b such that ℛ 1 and ℛ 2, sweep out the same volume when rotated about the x-axis?

(d) Is there a value of b such that ℛ 1, and ℛ 2, sweep out the same volume when rotated about the y-axis?

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Chapter 7 Solutions

Single Variable Essential Calculus: Early Transcendentals

Ch. 7.1 - Find the area of the shaded region.Ch. 7.1 - Prob. 2ECh. 7.1 - Find the area of the shaded region.Ch. 7.1 - Find the area of the shaded region.Ch. 7.1 - Prob. 5ECh. 7.1 - Sketch the region enclosed by the given curves....Ch. 7.1 - Prob. 7ECh. 7.1 - Sketch the region enclosed by the given curves....Ch. 7.1 - Prob. 9ECh. 7.1 - Prob. 10E
Ch. 7.1 - Sketch the region enclosed by the given curves and...Ch. 7.1 - 14. Sketch the region enclosed by the given curves...Ch. 7.1 - Sketch the region enclosed by the given curves and...Ch. 7.1 - Prob. 14ECh. 7.1 - Sketch the region enclosed by the given curves and...Ch. 7.1 - Sketch the region enclosed by the given curves and...Ch. 7.1 - Prob. 16ECh. 7.1 - Prob. 18ECh. 7.1 - Prob. 19ECh. 7.1 - Sketch the region enclosed by the given curves and...Ch. 7.1 - Sketch the region that lies between the curves y =...Ch. 7.1 - Graph the curves y = x2 x and y = x3 4x2 +3x on...Ch. 7.1 - Graph the region between the curves and use your...Ch. 7.1 - Graph the region between the curves and use your...Ch. 7.1 - Graph the region between the curves and use your...Ch. 7.1 - Graph the region between the curves and use your...Ch. 7.1 - Sketch the region in the xy-plane defined by the...Ch. 7.1 - Racing cars driven by Chris and Kelly are side by...Ch. 7.1 - The widths (in meters) of a kidney-shaped swimming...Ch. 7.1 - A cross-section of an airplane wing is shown....Ch. 7.1 - If the birth rate of a population is b(t) =...Ch. 7.1 - Two cars, A and B, start side by side and...Ch. 7.1 - Find a positive continuous function f such that...Ch. 7.1 - Find the area of the region bounded by the...Ch. 7.1 - Find the number b such that the line y = b divides...Ch. 7.1 - (a) Find the number a such that the line x = a...Ch. 7.1 - 59. Find the values of c such that the area of the...Ch. 7.1 - Suppose that 0 c /2. For what value of c is the...Ch. 7.1 - 61. For what values of m do the line y = mx and...Ch. 7.1 - 43. Find the area of the crescent-shaped region...Ch. 7.1 - A water storage tank has the shape of a cylinder...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - 1–12 Find the volume of the solid obtained by...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - Find the volume of the solid obtained by rotating...Ch. 7.2 - The region enclosed by the given curves is rotated...Ch. 7.2 - The region enclosed by the given curves is rotated...Ch. 7.2 - The region enclosed by the given curves is rotated...Ch. 7.2 - The region enclosed by the given curves is rotated...Ch. 7.2 - The region enclosed by the given curves is rotated...Ch. 7.2 - The region enclosed by the given curves is rotated...Ch. 7.2 - Set up an integral for the volume of the solid...Ch. 7.2 - Set up an integral for the volume of the solid...Ch. 7.2 - Set up an integral for the volume of the solid...Ch. 7.2 - Set up an integral for the volume of the solid...Ch. 7.2 - Use a graph to find approximate x-coordinates of...Ch. 7.2 - Use a graph to find approximate x-coordinates of...Ch. 7.2 - Each integral represents the volume of a solid....Ch. 7.2 - Each integral represents the volume of a solid....Ch. 7.2 - A CAT scan produces equally spaced cross-sectional...Ch. 7.2 - A log 10 m long is cut at 1-meter intervals and...Ch. 7.2 - 47. Find the volume of the described solid S. A...Ch. 7.2 - Find the volume of the described solid S. A...Ch. 7.2 - Find the volume of the described solid S. A cap of...Ch. 7.2 - Find the volume of the described solid S. A...Ch. 7.2 - Find the volume of the described solid S. A...Ch. 7.2 - Find the volume of the described solid S. A...Ch. 7.2 - Find the volume of the described solid S. A...Ch. 7.2 - 54. Find the volume of the described solid S. The...Ch. 7.2 - Find the volume of the described solid S. The base...Ch. 7.2 - Find the volume of the described solid S. The base...Ch. 7.2 - Find the volume of the described solid S. The base...Ch. 7.2 - Find the volume of the described solid S. The base...Ch. 7.2 - The base of S is the same base as in Exercise 42,...Ch. 7.2 - The base of S is a circular disk with radius r....Ch. 7.2 - (a) Set up an integral for the volume of a solid...Ch. 7.2 - A wedge is cut out of a circular cylinder of...Ch. 7.2 - (a) Cavalieris Principle states that if a family...Ch. 7.2 - Find the volume common to two circular cylinders,...Ch. 7.2 - Prob. 51ECh. 7.2 - A bowl is shaped like a hemisphere with diameter...Ch. 7.2 - A hole of radius r is bored through the middle of...Ch. 7.2 - A hole of radius r is bored through the center of...Ch. 7.2 - Some of the pioneers of calculus, such as Kepler...Ch. 7.3 - Let S be the solid obtained by rotating the region...Ch. 7.3 - Let S be the solid obtained by rotating the region...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Let V be the volume of the solid obtained by...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Prob. 14ECh. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - 17. Use the method of cylindrical shells to find...Ch. 7.3 - Use the method of cylindrical shells to find the...Ch. 7.3 - 15-20 Use the method of cylindrical shells to find...Ch. 7.3 - 15-20 Use the method of cylindrical shells to find...Ch. 7.3 - (a) Set up an integral for the volume of the solid...Ch. 7.3 - (a) Set up an integral for the volume of the solid...Ch. 7.3 - (a) Set up an integral for the volume of the solid...Ch. 7.3 - Prob. 24ECh. 7.3 - Prob. 25ECh. 7.3 - Prob. 26ECh. 7.3 - Use the Midpoint Rule with n = 4 to estimate the...Ch. 7.3 - Prob. 28ECh. 7.3 - Prob. 29ECh. 7.3 - Prob. 30ECh. 7.3 - Prob. 31ECh. 7.3 - Prob. 32ECh. 7.3 - Prob. 33ECh. 7.3 - Prob. 34ECh. 7.3 - Prob. 35ECh. 7.3 - Prob. 36ECh. 7.3 - Prob. 37ECh. 7.3 - Prob. 38ECh. 7.3 - Prob. 39ECh. 7.3 - 39-41 Use cylindrical shells to find the volume of...Ch. 7.3 - Use cylindrical shells to find the volume of the...Ch. 7.3 - Use cylindrical shells to find the volume of the...Ch. 7.3 - Prob. 43ECh. 7.4 - Use the arc length formula (3) to find the length...Ch. 7.4 - Use the arc length formula to find the length of...Ch. 7.4 - Set up an integral that represents the length of...Ch. 7.4 - Prob. 4ECh. 7.4 - Prob. 6ECh. 7.4 - Prob. 5ECh. 7.4 - Prob. 7ECh. 7.4 - Prob. 8ECh. 7.4 - Prob. 9ECh. 7.4 - Prob. 10ECh. 7.4 - Prob. 11ECh. 7.4 - Prob. 12ECh. 7.4 - Find the exact length of the curve. 15. y = ln(sec...Ch. 7.4 - Prob. 14ECh. 7.4 - Prob. 15ECh. 7.4 - Prob. 16ECh. 7.4 - Prob. 17ECh. 7.4 - Prob. 18ECh. 7.4 - Prob. 19ECh. 7.4 - Graph the curve and visually estimate its length....Ch. 7.4 - Prob. 21ECh. 7.4 - Prob. 22ECh. 7.4 - Prob. 23ECh. 7.4 - Use Simpsons Rule with n = 10 to estimate the arc...Ch. 7.4 - Prob. 25ECh. 7.4 - Prob. 27ECh. 7.4 - Prob. 28ECh. 7.4 - Prob. 29ECh. 7.4 - Prob. 30ECh. 7.4 - Find the arc length function for the curve...Ch. 7.4 - Prob. 32ECh. 7.4 - Prob. 33ECh. 7.4 - Prob. 35ECh. 7.4 - Prob. 34ECh. 7.4 - Prob. 36ECh. 7.5 - 41550-8.2-1E 1–6. (a) Set up an integral for the...Ch. 7.5 - Prob. 2ECh. 7.5 - Prob. 3ECh. 7.5 - Prob. 4ECh. 7.5 - Prob. 5ECh. 7.5 - Prob. 6ECh. 7.5 - 5-12 Find the exact area of the surface obtained...Ch. 7.5 - Prob. 8ECh. 7.5 - Prob. 9ECh. 7.5 - Prob. 10ECh. 7.5 - Prob. 11ECh. 7.5 - Find the exact area of the surface obtained by...Ch. 7.5 - Prob. 13ECh. 7.5 - Prob. 14ECh. 7.5 - Prob. 15ECh. 7.5 - The given curve is rotated about the y-axis. Find...Ch. 7.5 - Prob. 19ECh. 7.5 - Prob. 20ECh. 7.5 - Prob. 21ECh. 7.5 - Prob. 22ECh. 7.5 - Prob. 23ECh. 7.5 - Prob. 24ECh. 7.5 - Prob. 25ECh. 7.5 - Prob. 26ECh. 7.6 - Prob. 1ECh. 7.6 - Prob. 2ECh. 7.6 - Shown is the graph of a force function (in...Ch. 7.6 - Prob. 4ECh. 7.6 - Prob. 5ECh. 7.6 - A spring has a natural length of 20 cm. If a 25-N...Ch. 7.6 - Suppose that 2 J of work is needed to stretch a...Ch. 7.6 - If 6 J of work is needed to stretch a spring from...Ch. 7.6 - Prob. 9ECh. 7.6 - Show how to approximate the required work by a...Ch. 7.6 - Prob. 10ECh. 7.6 - Show how to approximate the required work by a...Ch. 7.6 - Show how to approximate the required work by a...Ch. 7.6 - Show how to approximate the required work by a...Ch. 7.6 - Show how to approximate the required work by a...Ch. 7.6 - 21. Show how to approximate the required work by a...Ch. 7.6 - The tank shown is full of water. Find the work...Ch. 7.6 - (a) The tank shown is full of water. Given that...Ch. 7.6 - 29. When gas expands in a cylinder with radius r,...Ch. 7.6 - Prob. 20ECh. 7.6 - 33. (a) Newton’s Law of Gravitation states that...Ch. 7.6 - The Great Pyramid of King Khufu was built of...Ch. 7.6 - Use an improper integral and information from...Ch. 7.6 - Find the escape velocity v0 that is needed to...Ch. 7.6 - 2529 A vertical plate is submerged (or partially...Ch. 7.6 - 2529 A vertical plate is submerged (or partially...Ch. 7.6 - A vertical plate is submerged (or partially...Ch. 7.6 - 2529 A vertical plate is submerged (or partially...Ch. 7.6 - A vertical plate is submerged (or partially...Ch. 7.6 - A milk truck carries milk with density 64.6 lb/ft3...Ch. 7.6 - A trough is filled with a liquid of density 840...Ch. 7.6 - A vertical dam has a semicircular gate as shown in...Ch. 7.6 - Prob. 32ECh. 7.6 - A swimming pool is 20 ft wide and 40 ft long and...Ch. 7.6 - A metal plate was found submerged vertically in...Ch. 7.6 - Point-masses mi arc located on the x-axis as...Ch. 7.6 - The masses mi, are located at the points Pi. Find...Ch. 7.6 - 41550-8.3-24E 23–24 The masses mi, are located at...Ch. 7.6 - Sketch the region bounded by the curves, and...Ch. 7.6 - Sketch the region bounded by the curves, and...Ch. 7.6 - Sketch the region bounded by the curves, and...Ch. 7.6 - Sketch the region bounded by the curves, and...Ch. 7.6 - Find the centroid of the region bounded by the...Ch. 7.6 - Find the centroid of the region bounded by the...Ch. 7.6 - Find the centroid of the region bounded by the...Ch. 7.6 - Find the centroid of the region bounded by the...Ch. 7.6 - 4748 Calculate the moments Mx and My and the...Ch. 7.6 - 4748 Calculate the moments Mx and My and the...Ch. 7.6 - Prove that the centroid of any triangle is located...Ch. 7.6 - Find the centroid of the region shown, not by...Ch. 7.6 - Find the centroid of the region shown, not by...Ch. 7.6 - Use the Theorem of Pappus to find the volume of...Ch. 7.6 - Use the Theorem of Pappus to find the volume of...Ch. 7.6 - Use the Theorem of Pappus to find the volume of...Ch. 7.6 - 41550-8.3-50E 50. Let ℛ be the region that lies...Ch. 7.6 - Prove Formulas 13. 13....Ch. 7.7 - Psychologists interested in learning theory study...Ch. 7.7 - Prob. 21ECh. 7.7 - Match the differential equation with its direction...Ch. 7.7 - Prob. 23ECh. 7.7 - Match the differential equation with its direction...Ch. 7.7 - Prob. 25ECh. 7.7 - Prob. 26ECh. 7.7 - 41550-9.2-9E 9–10 Sketch a direction field for the...Ch. 7.7 - Sketch a direction field for the differential...Ch. 7.7 - Sketch the direction field of the differential...Ch. 7.7 - Sketch the direction field of the differential...Ch. 7.7 - Sketch the direction field of the differential...Ch. 7.7 - Sketch the direction field of the differential...Ch. 7.7 - Prob. 1ECh. 7.7 - Prob. 2ECh. 7.7 - Solve the differential equation. 3. xyy=x2+1Ch. 7.7 - Solve the differential equation. (y2 + xy2)y = 1Ch. 7.7 - Solve the differential equation. (y + sin y)y = x...Ch. 7.7 - Solve the differential equation. dyd=eysin2ysecCh. 7.7 - Prob. 7ECh. 7.7 - Prob. 8ECh. 7.7 - Find the solution of the differential equation...Ch. 7.7 - Prob. 10ECh. 7.7 - Prob. 11ECh. 7.7 - Find the solution of the differential equation...Ch. 7.7 - Prob. 13ECh. 7.7 - Prob. 14ECh. 7.7 - Prob. 15ECh. 7.7 - Prob. 16ECh. 7.7 - Prob. 17ECh. 7.7 - Prob. 18ECh. 7.7 - A sphere with radius 1 m has temperature 15C. It...Ch. 7.7 - A glucose solution is administered intravenously...Ch. 7.7 - A certain small country has 10 billion in paper...Ch. 7.7 - Write the solution of the logistic initial-value...Ch. 7.7 - A tank contains 1000 L of brine with 15 kg of...Ch. 7.7 - The air in a room with volume 180 m3 contains...Ch. 7.7 - A vat with 500 gallons of beer contains 4% alcohol...Ch. 7.7 - A tank contains 1000 L of pure water. Brine that...Ch. 7.7 - When a raindrop falls, it increases in size and so...Ch. 7.7 - 41550-9.3-50E 50. An object of mass m is moving...Ch. 7.7 - According to Newtons Law of Universal Gravitation,...Ch. 7.7 - The Pacific halibut fishery has been modeled by...Ch. 7.7 - One model for the spread of a rumor is that the...Ch. 7.7 - 41550-9.4-12E 12. Biologists stocked a lake with...Ch. 7.7 - (a) Show that if y satisfies the logistic equation...Ch. 7.7 - For a fixed value of M (say M = 10), the family of...Ch. 7 - Prob. 1RCCCh. 7 - Suppose that Sue runs faster than Kathy throughout...Ch. 7 - 3. (a) Suppose S is a solid with known...Ch. 7 - (a) What is the volume of a cylindrical shell? (b)...Ch. 7 - Suppose that you push a book across a 6-meter-long...Ch. 7 - 1. Find the area of the region bounded by the...Ch. 7 - Prob. 2RECh. 7 - Prob. 4RECh. 7 - Prob. 3RECh. 7 - Prob. 5RECh. 7 - Prob. 6RECh. 7 - Prob. 7RECh. 7 - Prob. 8RECh. 7 - Prob. 9RECh. 7 - Prob. 10RECh. 7 - Prob. 11RECh. 7 - Prob. 12RECh. 7 - Prob. 13RECh. 7 - Prob. 14RECh. 7 - Prob. 15RECh. 7 - Prob. 16RECh. 7 - Prob. 17RECh. 7 - Prob. 18RECh. 7 - Prob. 19RECh. 7 - Prob. 20RECh. 7 - Prob. 21RECh. 7 - Prob. 22RECh. 7 - Prob. 23RECh. 7 - Prob. 24RECh. 7 - A force of 30 N is required to maintain a spring...Ch. 7 - A 1600-lb elevator is suspended by a 200-ft cable...Ch. 7 - 29. A tank full of water has the shape of a...Ch. 7 - Let 1, be the region bounded by y = x2, y = 0, and...Ch. 7 - Prob. 5RCCCh. 7 - Prob. 6RCCCh. 7 - Prob. 8RCCCh. 7 - Prob. 9RCCCh. 7 - Prob. 10RCCCh. 7 - Prob. 25RECh. 7 - Prob. 26RECh. 7 - Prob. 27RECh. 7 - Prob. 28RECh. 7 - Use Simpson's Rule with n = 10 to estimate the...Ch. 7 - Use Simpsons Rule with n = 10 to estimate the area...Ch. 7 - Find the length of the curve y=1xt1dt1x16Ch. 7 - Prob. 32RECh. 7 - A gate in an irrigation canal is constructed in...Ch. 7 - A trough is filled with water and its vertical...Ch. 7 - Find the centroid of the region shown.Ch. 7 - Find the centroid of the region bounded by the...Ch. 7 - Find the centroid of the region bounded by the...Ch. 7 - Find the volume obtained when the circle of radius...Ch. 7 - 41550-8-18RE 18. Use the Theorem of Pappus and the...Ch. 7 - Prob. 11RCCCh. 7 - Prob. 12RCCCh. 7 - Prob. 13RCCCh. 7 - (a) Sketch a direction field for the differential...Ch. 7 - Prob. 44RECh. 7 - Solve the differential equation. 7. 2yey2y=2x+3xCh. 7 - Solve the initial-value problem. 9....Ch. 7 - 41550-9-10RE 9–11. Solve the initial-value...
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