   Chapter 15.1, Problem 43E

Chapter
Section
Textbook Problem

Find the volume of the solid enclosed by the paraboloid z = 2 + x2 + (y − 2)2 and the planes z = 1, x = 1, x = −1, y = 0, and y = 4.

To determine

To find: The volume of the solid in the first octant bounded by the cylinder.

Explanation

Formula used:

The volume of the solid, V=RzdA , where, z is the given function.

Given:

The surface is z=2+x2+(y2)2 .

The planes are z=1,x=1,x=1&y=0,y=4 .

Calculation:

The rectangular region is R=[1,1]×[0,4] and the solid lies below z=1 .

Therefore, the volume of the solid is obtained as follows.

V=RzdAR1dA=1104(2+x2+(y2)2)dydx1104(1)dydx

First, compute the integral with respect to y.

V=11[2y+x2y+(y2)33]04dx11[y]04dx

Apply the limit value for y,

V=11[(4x2+2

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