find the volume of the solid that is bounded by y2z29 x0y3x

Solution

Here are the steps. 1. Solve the equation for z to get z = square root of [9 - y^2]. This is your integrand in the double integral. 2. The region you want to integrate over is a triangle in the x/y plane bounded by the lines y = 3, x = 0, y = 3x. 3. It is easiest to do dxdy since then you can factor the integrand right out of the first (inside) integral. Here, on the inside integral, you have x goes from 0 to y/3. On the outside integral you have y go from 0 to 3. 4. Do the integration to obtain the final answer.