   Chapter 15, Problem 31RE

Chapter
Section
Textbook Problem

Calculate the value of the multiple integral.31. ∭ E y 2 z 2   d V , where E is bounded by the paraboloid x = 1 − y2 − z2 and the plane x = 0

To determine

To calculate: The given triple integral.

Explanation

Given:

The function is f(x,y,z)=y2z2 .

The region E is bounded by the paraboloid x=1y2z2 and the plane x=0 .

Formula used:

If f is a polar rectangle R given by 0arb,αθβ, where 0βα2π , then, Rf(x,y)dA=αβabf(rcosθ,rsinθ)rdrdθ (1)

If g(x) is the function of x and h(y) is the function of y then,

abcdg(x)h(y)dydx=abg(x)dxcdh(y)dy (2)

Calculation:

Substitute x=0 in the given equation as shown below.

(0)=1y2z2z2=y21z2=1y2z=±1y2

From the given conditions, x varies from 0 to 1y2z2 , y varies from 1 to 1 and z varies from 1y2 to 1y2 . First compute the integral with respect to z and apply the limit.

Ey2z2dV=111y21y201y2z2y2z2dxdzdy=111y21y2[y2z2x]01y2z2dzdy=111y21y2[y2z2(1y2z20)]dzdy=111y21y2[y2z2(1y2z2)]dzdy

Since the value to be integrated and the limit of z are difficult, use of polar coordinates makes the problem easier. Substitute y=rcosθ,z=rsinθ

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