   Chapter 15.6, Problem 11E

Chapter
Section
Textbook Problem

Evaluate the triple integral.11. ∭ E z x 2   +   z 2 ​   d V ,   where E   =   { ( x ,   y ,   z ) |   1   ≤   y   ≤   4 ,       y   ≤   z   ≤   4 ,     0   ≤   x     ≤   z }

To determine

To evaluate: The given triple integral.

Explanation

Given:

The function is f(x,y,z)=zx2+z2 .

The region is E={(x,y,z)|0xz,1y4,yz4} .

Calculation:

The given integral is, Ezx2+z2dV=14y40zzx2+z2dxdzdy .

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

Ezx2+z2dV=14y40zzx2+z2dxdzdy=14y4z[1ztan1(xz)]0zdzdy[dxa2+x2=1atan1xa+c]=14y4z[(1ztan1(zz))(1ztan1(0z))]dzdy

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