Question

Find the volume of the solid in the first octant bounded by the coordinate planes, the cylinder X^2+Y^2=4 and the plane X+Z=4.

Answer 1

`4\pi- (8)/(3)`

Explanation:

The solid that you described is shown in the image of below. To compute the volume we use triple integrals. Observe that

`0\leq x\leq 2`

`0\leq y \leq √(4-x^2)`

`0\leq z \leq 4-x`

So, the volumen of the solid is given by

`\int_(0)^(2)\int_(0)^(√(4-x^2))\int_(0)^(4-x)1\, dzdydx=\int_(0)^(2)\int_(0)^(√(4-x^2))4-x\, dydx=\int_(0)^(2)(4-x)√(4-x^2)\,dx`

This last integral equals
`4\pi -(8)/(3)`