Question

Evaluate [(triple integral) in E] xy dV, where E is the solid tetrahedron with vertices (0, 0, 0), (1, 0,0) , (0,2,0) and (0,0,3)

Answer 1

3

Explanation:

The given points indicate that the triple integral should be from 0-1 in the x-direction, 0-2 in the y-direction, and 0-3 in the z-direction. So,

`\int\limits^3_0 \int\limits^2_0\int\limits^1_0 {xy} \, dxdydz`

This triple integral can be broken into three steps. Evaluate the innermost integral with respect to x:

`\int\limits^1_0 {xy} \, dx\\(1)/(2)x^2y]_0^1\\(1)/(2)[(1)^2y]-(1)/(2)[(0)^2y]\\(1)/(2)y`

Next, use
`(1)/(2)y` to evaluate the middle integral with respect to y:

`\int\limits^2_0 {(1)/(2)y} \, dy\\(1)/(2)\int\limits^2_0 {y} \, dy\\(1)/(2)((1)/(2)y^2)]_0^2\\(1)/(4)[(2)^2-(0)^2]\\(1)/(4)(4)\\1`

Finally, use 1 to evaluate the outermost integral with respect to z:

`\int\limits^3_0 {1} \, dz\\z]_0^3\\(3)-(0)\\3`

So, the final value of the triple integral is 3