Question

Find the volume of the solid under the surface z=xy and above the triangular region in the xy-plane bounded by the lines y=2x, y=-x+6, and y=0.

Answer 1

@wio @radar @robtobey @Directrix

Answer 2

draw it

Answer 3

2x=-x+6
x=2
they intersect at x=2

Answer 4

Okay, I think that we can start with :\[
0\leq z \leq xy
\]

Answer 5

I think \(y=0\) means it is \(y\) simple.

Answer 6

So what are the limits of integration for x and y?

Answer 7

First solve for \(x\) in terms of \(y\).

Answer 8

We have \[
x=\frac y2\\x=6-y
\]We know at \(0\), that the second equation will have a greater \(x\), so we say: \[
\frac y2 \leq x \leq 6-y
\]

Answer 9

Finally, we know \(y\) starts at \(0\), and solving for the intersection, we get \(y=4\).\[
0\leq y \leq 4
\]

Answer 10

Thank you!