   Chapter 15.6, Problem 22E

Chapter
Section
Textbook Problem

Use a triple integral to find the volume of the given solid.22. The solid enclosed by the cylinder x2 + z2 = 4 and the planes y = -1 and y + z = 4

To determine

To find: The volume of the solid enclosed by the cylinder x2+z2=4 and the planes y=1 and y+z=4 by using the triple integral.

Explanation

Cylinder and planes intersect each other to form the required solid.

Express the surface of the cylinder x2+z2=4 in polar form by using polar coordinates. Let x=rcosθ and z=rsinθ.

The surface of the cylinder is in the circle form. So, limit of θ varies from 0 to 2π and the limit of r varies from 0 to 2.

From the given planes, y varies from 1 to 4rsinθ, since z=rsinθ.

Let E be the region enclosed by solid where E={(θ,r,y)|0θ2π,0r2,1y4rsinθ}.

The volume of integral for the solid is, EdV=02π0214rsinθdydrdθ.

Integrate the given integral with respect to y and apply the limit of it.

EdV=02π02[y]14rsinθrdrdθ=02π02[4rsinθ(1)]rdrdθ=02π02[4rsinθ+1]rdrdθ=02π02[5rr2sinθ]drdθ

Integrate the given integral with respect to r and apply the limit of it

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