Volume of the solid that lies under $x+y+z=10$ and above the triangular region $R$ that is bounded by the y-axis and the lines $y=5$, $y=x$.

Question

Volume of the solid that lies under $x+y+z=10$ and above the triangular region $R$ that is bounded by the y-axis and the lines $y=5$, $y=x$.

I've been stuck on this question for a while, I know its gonna be a double integral I just can't get the boundaries.

Answer 1

Firstly draw out $R$.

This will help you think about the bounds of your integral. This will also help you decide if it is better to integrate over $x$ first or $y$.

There are two ways to write the integral (based on which variable is integrated first):

$$\int_0^5\int_x^5 10-x-y dydx$$

OR

$$\int_0^5\int_0^y 10-x-y dxdy$$