Explanation Given: The equation of three cylinders: x 2 + z 2 = 1 , y 2 + z 2 = 1 and x 2 + y 2 = 1 . Formula used: The polar coordinates ( r , θ ) are related to the rectangular coordinates ( x , y ) by: x = r cos θ and y = r sin θ . Calculation: First of all look at the intersection of two cylinders at x- axis and y- axis. It can be seen that there are four equal parts The third cylinder x 2 + y 2 = 1 appears as a circle. To calculate the volumejust look at one fourth of the body where z > 0 . The radius goes from 0 to a , the angle from − π 4 to π 4 . The equation of the top of the body is z = 1 − x 2 as in this region just the cylinder x 2 + z 2 = 1 matters. Put x = r cos t and y = r sin t to change the equation in polar form. So, volume of that one part is given by ∫ − π 4 π 4 ∫ 0 1 1 − r 2 cos 2 ( t ) r d r d t . Now, the volume of the solid will be eight times of that volume

11th Edition

ISBN: 9781337275347

Author: Ron Larson, Bruce H. Edwards

Publisher: Cengage Learning

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Chapter 14, Problem 1PS

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To calculate: The volume of the solid of intersection of three cylinders x2+z2=1, y2+z2=1 and x2+y2=1.

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Chapter 14, Problem 2PS

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The volume of the solid of intersection of three cylinders x 2 + z 2 = 1 , y 2 + z 2 = 1 and x 2 + y 2 = 1 .

Solution Summary: The author calculates the volume of the intersection of three cylinders: 4.6863 cubic unit.

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To calculate: The volume of the solid of intersection of three cylinders x2+z2=1, y2+z2=1 and x2+y2=1.

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Chapter 14 Solutions