   Chapter 15.6, Problem 15E

Chapter
Section
Textbook Problem

Evaluate the triple integral.15. ∭ T y 2   d V . where T is the solid tetrahedron with vertices (0, 0,0), (2, 0, 0). (0, 2, 0). and (0, 0, 2)

To determine

To evaluate: The given triple integral.

Explanation

Given:

The function is f(x,y,z)=y2 .

The region T is the tetrahedron with vertices, (0,0,0),(2,0,0),(0,2,0),(0,0,2) .

Calculation:

From the given conditions, it is observed that the equations of the sides of the tetrahedron will be 2x and 2xy . Hence, T={(x,y,z)|0x2,0y2x,0z2xy} . Thus, the given integral is, Ty2dV=0202x02xyy2dzdydx .

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

Ty2dV=0202xy2[z]02xydydx=0202xy2[2xy0]dydx=0202x(2y2xy2y3)dydx

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

Ty2dV=02[2y33xy33y44]02xdx=02[(2(2x)33x(2x)33(2x)44)(2(0)33x(0)33(0)4

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