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.

Question

idealist10 · Answer

@wio @radar @robtobey @Directrix

anonymous · Answer

draw it

idealist10 · Answer

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

anonymous · Answer

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

anonymous · Answer

I think $y=0$ means it is $y$ simple.

idealist10 · Answer

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

anonymous · Answer

First solve for $x$ in terms of $y$.

anonymous · Answer

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

anonymous · Answer

Finally, we know $y$ starts at $0$, and solving for the intersection, we get $y=4$.$$
0\leq y \leq 4
$$

idealist10 · Answer

Thank you!