   Chapter 15.6, Problem 7E

Chapter
Section
Textbook Problem

Evaluate the iterated integral.7. ∫ 0 π ∫ 0 1 ∫ 0 1   − z 2   z   sin   x   d y   d z   d x

To determine

To evaluate: The iterated integral.

Explanation

Given:

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

The region is B={(x,y,z)|0xπ,0y1z2,0z1} .

Formula used:

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 (1)

Calculation:

The given integral is, 0π0101z2zsinxdydzdx .

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

0π0101z2zsinxdydzdx=0π01zsinx[y]01z2dzdx=0π01zsinx[1z20]dzdx=0π01zsinx1z2dzdx

Use the equation (1), 0π0101z2zsinxdydzdx=0πsinxdx01z1z2dz (2)

By substitution method, let u=z2 .

Then, the derivative of u is du=2dz .

Thus, there is a change in the limit values of the integration such as,

If z=0 , then u=0 and if z=1 , then u=1

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