   Chapter 15.2, Problem 26E

Chapter
Section
Textbook Problem

Find the volume of the given solid.26. Enclosed by the paraboloid z = x2 + y2 + 1 and the planes x = 0, y = 0, z = 0, and x + y = 2

To determine

To find: The volume of the solid that enclosed by the paraboloid and the planes.

Explanation

Given:

The paraboloid is z=x2+y2+1 .

The planes are, x=0,y=0,z=0,x+y=2 .

Formula used:

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

Calculation:

The plane equation can be modified as y=2x and it is observed from the given equations of planes that y varies from 0 to 2x and x varies from 0 to 2. Thus, the volume of the solid is computed as follows.

V=RzdA=0202x(x2+y2+1)dydx

First, compute the integral with respect to y.

V=02[x2y+y33+y]02xdx

Apply the limit value for y,

V=02[(x2(2x)+(2x)33+(2x))(0+0+0)]dx=02[2x2x3+8312x3+6x23x33+2x]dx=02[4x33+4x25x+143]dx

Compute the integral with respect to x

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