   Chapter 15.6, Problem 16E

Chapter
Section
Textbook Problem

Evaluate the triple integral.16. ∭ T x z   d V , where T is the solid tetrahedron with vertices (0, 0, 0), (1, 0, 1), (0, 1, 1), and (0, 0, 1)

To determine

To evaluate: The given triple integral.

Explanation

Given:

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

The region T is the tetrahedron with vertices (0,0,0),(1,0,1),(0,1,1),(0,0,1) .

Calculation:

From the given conditions, it is observed that the equations of the sides of the tetrahedron are x and zx . Hence, T={(x,y,z)|0x1,0yzx,xz1} . Thus, the given integral is, TxzdV=01x10zxxzdydzdx .

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

TxzdV=01x1xz[y]0zxdzdx=01x1xz[zx0]dzdx=01x1xz[zx]dzdx=01x1[xz2x2z]dzdx

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

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