Explanation Given : The region bounded by the two paraboloids z = x 2 + y 2 and z = 8 − x 2 − y 2 . Given the region bounded by by the two paraboloids z = x 2 + y 2 and z = 8 − x 2 − y 2 . First we find the intersection of the two surfaces as it bounds the variation of x and y coordinates of the points inside this volume. Hence, consider the following x 2 + y 2 = 8 − x 2 − y 2 2 x 2 + 2 y 2 = 8 x 2 + y 2 = 4 and hence all points (x,y,z) which lie on both surfaces satisfy { x 2 + y 2 = 4 z = x 2 + y 2 o r { x 2 + y 2 = 4 z = 8 − x 2 − y 2 which are two different equations of the same intersection curve. Now, we are ready to describe the coordinates of the points inside the volume in some mathematical language.Therefore the boundary of projected region R in the x-y plane is given by the circle x 2 + y 2 = 4 ...

3rd Edition

ISBN: 9781464125263

Author: Jon Rogawski, Colin Adams

Publisher: W.H. Freeman & Co

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Chapter 16.2, Problem 49E

To determine

The double integral that gives the volume of the region bounded by the two paraboloids

$z = x^{2} + y^{2}$ and $z = 8 - x^{2} - y^{2}$ .

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Chapter 16.2, Problem 48E

Chapter 16.2, Problem 50E

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Chapter 16 Solutions

Calculus - Standalone book

Ch. 16.1 - Prob. 1PQ Ch. 16.1 - Prob. 2PQ Ch. 16.1 - Prob. 3PQ Ch. 16.1 - Prob. 4PQ Ch. 16.1 - Prob. 5PQ Ch. 16.1 - Prob. 6PQ Ch. 16.1 - Prob. 1E Ch. 16.1 - Prob. 2E Ch. 16.1 - Prob. 3E Ch. 16.1 - Prob. 4E

Ch. 16.1 - Prob. 5E Ch. 16.1 - Prob. 6E Ch. 16.1 - Prob. 7E Ch. 16.1 - Prob. 8E Ch. 16.1 - Prob. 9E Ch. 16.1 - Prob. 10E Ch. 16.1 - Prob. 11E Ch. 16.1 - Prob. 12E Ch. 16.1 - Prob. 13E Ch. 16.1 - Prob. 14E Ch. 16.1 - Prob. 15E Ch. 16.1 - Prob. 16E Ch. 16.1 - Prob. 17E Ch. 16.1 - Prob. 18E Ch. 16.1 - Prob. 19E Ch. 16.1 - Prob. 20E Ch. 16.1 - Prob. 21E Ch. 16.1 - Prob. 22E Ch. 16.1 - Prob. 23E Ch. 16.1 - Prob. 24E Ch. 16.1 - Prob. 25E Ch. 16.1 - Prob. 26E Ch. 16.1 - Prob. 27E Ch. 16.1 - Prob. 28E Ch. 16.1 - Prob. 29E Ch. 16.1 - Prob. 30E Ch. 16.1 - Prob. 31E Ch. 16.1 - Prob. 32E Ch. 16.1 - Prob. 33E Ch. 16.1 - Prob. 34E Ch. 16.1 - Prob. 35E Ch. 16.1 - Prob. 36E Ch. 16.1 - Prob. 37E Ch. 16.1 - Prob. 38E Ch. 16.1 - Prob. 39E Ch. 16.1 - Prob. 40E Ch. 16.1 - Prob. 41E Ch. 16.1 - Prob. 42E Ch. 16.1 - Prob. 43E Ch. 16.1 - Prob. 44E Ch. 16.1 - Prob. 45E Ch. 16.1 - Prob. 46E Ch. 16.1 - Prob. 47E Ch. 16.1 - Prob. 48E Ch. 16.1 - Prob. 49E Ch. 16.1 - Prob. 50E Ch. 16.1 - Prob. 51E Ch. 16.1 - Prob. 52E Ch. 16.1 - Prob. 53E Ch. 16.2 - Prob. 1PQ Ch. 16.2 - Prob. 2PQ Ch. 16.2 - Prob. 3PQ Ch. 16.2 - Prob. 4PQ Ch. 16.2 - Prob. 1E Ch. 16.2 - Prob. 2E Ch. 16.2 - Prob. 3E Ch. 16.2 - Prob. 4E Ch. 16.2 - Prob. 5E Ch. 16.2 - Prob. 6E Ch. 16.2 - Prob. 7E Ch. 16.2 - Prob. 8E Ch. 16.2 - Prob. 9E Ch. 16.2 - Prob. 10E Ch. 16.2 - Prob. 11E Ch. 16.2 - Prob. 12E Ch. 16.2 - Prob. 13E Ch. 16.2 - Prob. 14E Ch. 16.2 - Prob. 15E Ch. 16.2 - Prob. 16E Ch. 16.2 - Prob. 17E Ch. 16.2 - Prob. 18E Ch. 16.2 - Prob. 19E Ch. 16.2 - Prob. 20E Ch. 16.2 - Prob. 21E Ch. 16.2 - Prob. 22E Ch. 16.2 - Prob. 23E Ch. 16.2 - Prob. 24E Ch. 16.2 - Prob. 25E Ch. 16.2 - Prob. 26E Ch. 16.2 - Prob. 27E Ch. 16.2 - Prob. 28E Ch. 16.2 - Prob. 29E Ch. 16.2 - Prob. 30E Ch. 16.2 - Prob. 31E Ch. 16.2 - Prob. 32E Ch. 16.2 - Prob. 33E Ch. 16.2 - Prob. 34E Ch. 16.2 - Prob. 35E Ch. 16.2 - Prob. 36E Ch. 16.2 - Prob. 37E Ch. 16.2 - Prob. 38E Ch. 16.2 - Prob. 39E Ch. 16.2 - Prob. 40E Ch. 16.2 - Prob. 41E Ch. 16.2 - Prob. 42E Ch. 16.2 - Prob. 43E Ch. 16.2 - Prob. 44E Ch. 16.2 - Prob. 45E Ch. 16.2 - Prob. 46E Ch. 16.2 - Prob. 47E Ch. 16.2 - Prob. 48E Ch. 16.2 - Prob. 49E Ch. 16.2 - Prob. 50E Ch. 16.2 - Prob. 51E Ch. 16.2 - Prob. 52E Ch. 16.2 - Prob. 53E Ch. 16.2 - Prob. 54E Ch. 16.2 - Prob. 55E Ch. 16.2 - Prob. 56E Ch. 16.2 - Prob. 57E Ch. 16.2 - Prob. 58E Ch. 16.2 - Prob. 59E Ch. 16.2 - Prob. 60E Ch. 16.2 - Prob. 61E Ch. 16.2 - Prob. 62E Ch. 16.2 - Prob. 63E Ch. 16.2 - Prob. 64E Ch. 16.2 - Prob. 65E Ch. 16.2 - Prob. 66E Ch. 16.2 - Prob. 67E Ch. 16.2 - Prob. 68E Ch. 16.3 - Prob. 1PQ Ch. 16.3 - Prob. 2PQ Ch. 16.3 - Prob. 3PQ Ch. 16.3 - Prob. 1E Ch. 16.3 - Prob. 2E Ch. 16.3 - Prob. 3E Ch. 16.3 - Prob. 4E Ch. 16.3 - Prob. 5E Ch. 16.3 - Prob. 6E Ch. 16.3 - Prob. 7E Ch. 16.3 - Prob. 8E Ch. 16.3 - Prob. 9E Ch. 16.3 - Prob. 10E Ch. 16.3 - Prob. 11E Ch. 16.3 - Prob. 12E Ch. 16.3 - Prob. 13E Ch. 16.3 - Prob. 14E Ch. 16.3 - Prob. 15E Ch. 16.3 - Prob. 16E Ch. 16.3 - Prob. 17E Ch. 16.3 - Prob. 18E Ch. 16.3 - Prob. 19E Ch. 16.3 - Prob. 20E Ch. 16.3 - Prob. 21E Ch. 16.3 - Prob. 22E Ch. 16.3 - Prob. 23E Ch. 16.3 - Prob. 24E Ch. 16.3 - Prob. 25E Ch. 16.3 - Prob. 26E Ch. 16.3 - Prob. 27E Ch. 16.3 - Prob. 28E Ch. 16.3 - Prob. 29E Ch. 16.3 - Prob. 30E Ch. 16.3 - Prob. 31E Ch. 16.3 - Prob. 32E Ch. 16.3 - Prob. 33E Ch. 16.3 - Prob. 34E Ch. 16.3 - Prob. 35E Ch. 16.3 - Prob. 36E Ch. 16.3 - Prob. 37E Ch. 16.3 - Prob. 38E Ch. 16.3 - Prob. 39E Ch. 16.3 - Prob. 40E Ch. 16.3 - Prob. 41E Ch. 16.3 - Prob. 42E Ch. 16.3 - Prob. 43E Ch. 16.3 - Prob. 44E Ch. 16.4 - Prob. 1PQ Ch. 16.4 - Prob. 2PQ Ch. 16.4 - Prob. 3PQ Ch. 16.4 - Prob. 4PQ Ch. 16.4 - Prob. 1E Ch. 16.4 - Prob. 2E Ch. 16.4 - Prob. 3E Ch. 16.4 - Prob. 4E Ch. 16.4 - Prob. 5E Ch. 16.4 - Prob. 6E Ch. 16.4 - Prob. 7E Ch. 16.4 - Prob. 8E Ch. 16.4 - Prob. 9E Ch. 16.4 - Prob. 10E Ch. 16.4 - Prob. 11E Ch. 16.4 - Prob. 12E Ch. 16.4 - Prob. 13E Ch. 16.4 - Prob. 14E Ch. 16.4 - Prob. 15E Ch. 16.4 - Prob. 16E Ch. 16.4 - Prob. 17E Ch. 16.4 - Prob. 18E Ch. 16.4 - Prob. 19E Ch. 16.4 - Prob. 20E Ch. 16.4 - Prob. 21E Ch. 16.4 - Prob. 22E Ch. 16.4 - Prob. 23E Ch. 16.4 - Prob. 24E Ch. 16.4 - Prob. 25E Ch. 16.4 - Prob. 26E Ch. 16.4 - Prob. 27E Ch. 16.4 - Prob. 28E Ch. 16.4 - Prob. 29E Ch. 16.4 - Prob. 30E Ch. 16.4 - Prob. 31E Ch. 16.4 - Prob. 32E Ch. 16.4 - Prob. 33E Ch. 16.4 - Prob. 34E Ch. 16.4 - Prob. 35E Ch. 16.4 - Prob. 36E Ch. 16.4 - Prob. 37E Ch. 16.4 - Prob. 38E Ch. 16.4 - Prob. 39E Ch. 16.4 - Prob. 40E Ch. 16.4 - Prob. 41E Ch. 16.4 - Prob. 42E Ch. 16.4 - Prob. 43E Ch. 16.4 - Prob. 44E Ch. 16.4 - Prob. 45E Ch. 16.4 - Prob. 46E Ch. 16.4 - Prob. 47E Ch. 16.4 - Prob. 48E Ch. 16.4 - Prob. 49E Ch. 16.4 - Prob. 50E Ch. 16.4 - Prob. 51E Ch. 16.4 - Prob. 52E Ch. 16.4 - Prob. 53E Ch. 16.4 - Prob. 54E Ch. 16.4 - Prob. 55E Ch. 16.4 - Prob. 56E Ch. 16.4 - Prob. 57E Ch. 16.4 - Prob. 58E Ch. 16.4 - Prob. 59E Ch. 16.4 - Prob. 60E Ch. 16.5 - Prob. 1PQ Ch. 16.5 - Prob. 2PQ Ch. 16.5 - Prob. 3PQ Ch. 16.5 - Prob. 1E Ch. 16.5 - Prob. 2E Ch. 16.5 - Prob. 3E Ch. 16.5 - Prob. 4E Ch. 16.5 - Prob. 5E Ch. 16.5 - Prob. 6E Ch. 16.5 - Prob. 7E Ch. 16.5 - Prob. 8E Ch. 16.5 - Prob. 9E Ch. 16.5 - Prob. 10E Ch. 16.5 - Prob. 11E Ch. 16.5 - Prob. 12E Ch. 16.5 - Prob. 13E Ch. 16.5 - Prob. 14E Ch. 16.5 - Prob. 15E Ch. 16.5 - Prob. 16E Ch. 16.5 - Prob. 17E Ch. 16.5 - Prob. 18E Ch. 16.5 - Prob. 19E Ch. 16.5 - Prob. 20E Ch. 16.5 - Prob. 21E Ch. 16.5 - Prob. 22E Ch. 16.5 - Prob. 23E Ch. 16.5 - Prob. 24E Ch. 16.5 - Prob. 25E Ch. 16.5 - Prob. 26E Ch. 16.5 - Prob. 27E Ch. 16.5 - Prob. 28E Ch. 16.5 - Prob. 29E Ch. 16.5 - Prob. 30E Ch. 16.5 - Prob. 31E Ch. 16.5 - Prob. 32E Ch. 16.5 - Prob. 33E Ch. 16.5 - Prob. 34E Ch. 16.5 - Prob. 35E Ch. 16.5 - Prob. 36E Ch. 16.5 - Prob. 37E Ch. 16.5 - Prob. 38E Ch. 16.5 - Prob. 39E Ch. 16.5 - Prob. 40E Ch. 16.5 - Prob. 41E Ch. 16.5 - Prob. 42E Ch. 16.5 - Prob. 43E Ch. 16.5 - Prob. 44E Ch. 16.5 - Prob. 45E Ch. 16.5 - Prob. 46E Ch. 16.5 - Prob. 47E Ch. 16.5 - Prob. 48E Ch. 16.5 - Prob. 49E Ch. 16.5 - Prob. 50E Ch. 16.5 - Prob. 51E Ch. 16.5 - Prob. 52E Ch. 16.5 - Prob. 53E Ch. 16.5 - Prob. 54E Ch. 16.5 - Prob. 55E Ch. 16.5 - Prob. 56E Ch. 16.5 - Prob. 57E Ch. 16.5 - Prob. 58E Ch. 16.5 - Prob. 59E Ch. 16.5 - Prob. 60E Ch. 16.5 - Prob. 61E Ch. 16.5 - Prob. 62E Ch. 16.5 - Prob. 63E Ch. 16.5 - Prob. 64E Ch. 16.6 - Prob. 1PQ Ch. 16.6 - Prob. 2PQ Ch. 16.6 - Prob. 3PQ Ch. 16.6 - Prob. 4PQ Ch. 16.6 - Prob. 1E Ch. 16.6 - Prob. 2E Ch. 16.6 - Prob. 3E Ch. 16.6 - Prob. 4E Ch. 16.6 - Prob. 5E Ch. 16.6 - Prob. 6E Ch. 16.6 - Prob. 7E Ch. 16.6 - Prob. 8E Ch. 16.6 - Prob. 9E Ch. 16.6 - Prob. 10E Ch. 16.6 - Prob. 11E Ch. 16.6 - Prob. 12E Ch. 16.6 - Prob. 13E Ch. 16.6 - Prob. 14E Ch. 16.6 - Prob. 15E Ch. 16.6 - Prob. 16E Ch. 16.6 - Prob. 17E Ch. 16.6 - Prob. 18E Ch. 16.6 - Prob. 19E Ch. 16.6 - Prob. 20E Ch. 16.6 - Prob. 21E Ch. 16.6 - Prob. 22E Ch. 16.6 - Prob. 23E Ch. 16.6 - Prob. 24E Ch. 16.6 - Prob. 25E Ch. 16.6 - Prob. 26E Ch. 16.6 - Prob. 27E Ch. 16.6 - Prob. 28E Ch. 16.6 - Prob. 29E Ch. 16.6 - Prob. 30E Ch. 16.6 - Prob. 31E Ch. 16.6 - Prob. 32E Ch. 16.6 - Prob. 33E Ch. 16.6 - Prob. 34E Ch. 16.6 - Prob. 35E Ch. 16.6 - Prob. 36E Ch. 16.6 - Prob. 37E Ch. 16.6 - Prob. 38E Ch. 16.6 - Prob. 39E Ch. 16.6 - Prob. 40E Ch. 16.6 - Prob. 41E Ch. 16.6 - Prob. 42E Ch. 16.6 - Prob. 43E Ch. 16.6 - Prob. 44E Ch. 16.6 - Prob. 45E Ch. 16.6 - Prob. 46E Ch. 16.6 - Prob. 47E Ch. 16.6 - Prob. 48E Ch. 16.6 - Prob. 49E Ch. 16.6 - Prob. 50E Ch. 16.6 - Prob. 51E Ch. 16.6 - Prob. 52E Ch. 16 - Prob. 1CRE Ch. 16 - Prob. 2CRE Ch. 16 - Prob. 3CRE Ch. 16 - Prob. 4CRE Ch. 16 - Prob. 5CRE Ch. 16 - Prob. 6CRE Ch. 16 - Prob. 7CRE Ch. 16 - Prob. 8CRE Ch. 16 - Prob. 9CRE Ch. 16 - Prob. 10CRE Ch. 16 - Prob. 11CRE Ch. 16 - Prob. 12CRE Ch. 16 - Prob. 13CRE Ch. 16 - Prob. 14CRE Ch. 16 - Prob. 15CRE Ch. 16 - Prob. 16CRE Ch. 16 - Prob. 17CRE Ch. 16 - Prob. 18CRE Ch. 16 - Prob. 19CRE Ch. 16 - Prob. 20CRE Ch. 16 - Prob. 21CRE Ch. 16 - Prob. 22CRE Ch. 16 - Prob. 23CRE Ch. 16 - Prob. 24CRE Ch. 16 - Prob. 25CRE Ch. 16 - Prob. 26CRE Ch. 16 - Prob. 27CRE Ch. 16 - Prob. 28CRE Ch. 16 - Prob. 29CRE Ch. 16 - Prob. 30CRE Ch. 16 - Prob. 31CRE Ch. 16 - Prob. 32CRE Ch. 16 - Prob. 33CRE Ch. 16 - Prob. 34CRE Ch. 16 - Prob. 35CRE Ch. 16 - Prob. 36CRE Ch. 16 - Prob. 37CRE Ch. 16 - Prob. 38CRE Ch. 16 - Prob. 39CRE Ch. 16 - Prob. 40CRE Ch. 16 - Prob. 41CRE Ch. 16 - Prob. 42CRE Ch. 16 - Prob. 43CRE Ch. 16 - Prob. 44CRE Ch. 16 - Prob. 45CRE Ch. 16 - Prob. 46CRE Ch. 16 - Prob. 47CRE Ch. 16 - Prob. 48CRE Ch. 16 - Prob. 49CRE Ch. 16 - Prob. 50CRE Ch. 16 - Prob. 51CRE Ch. 16 - Prob. 52CRE Ch. 16 - Prob. 53CRE Ch. 16 - Prob. 54CRE Ch. 16 - Prob. 55CRE Ch. 16 - Prob. 56CRE Ch. 16 - Prob. 57CRE Ch. 16 - Prob. 58CRE Ch. 16 - Prob. 59CRE Ch. 16 - Prob. 60CRE Ch. 16 - Prob. 61CRE

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The double integral that gives the volume of the region bounded by the two paraboloids z = x 2 + y 2 and z = 8 − x 2 − y 2 .

Solution Summary: The author explains that the double integral that gives the volume of the region bounded by the two paraboloids z = x 2 + y 2.

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The double integral that gives the volume of the region bounded by the two paraboloids

$z = x^{2} + y^{2}$ and $z = 8 - x^{2} - y^{2}$ .

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Chapter 16 Solutions

The double integral that gives the volume of the region bounded by the two paraboloids z = x 2 + y 2 and z = 8 − x 2 − y 2 .

Solution Summary: The author explains that the double integral that gives the volume of the region bounded by the two paraboloids z = x 2 + y 2.

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The double integral that gives the volume of the region bounded by the two paraboloids z=x2+y2 and z=8−x2−y2.

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Chapter 16 Solutions

The double integral that gives the volume of the region bounded by the two paraboloids

$z = x^{2} + y^{2}$ and $z = 8 - x^{2} - y^{2}$ .