What Is the Correct Approach to Solve This Double Integration Problem?

[Enzo]

Homework Statement

<br /> \int^1_y\int^1_0 x^2*e^{xy} dydx<br />Answer: 1/2 (e-2)

The Attempt at a Solution

I've tried about 4 ways of doing this, I can't solve it. It either ends up being a completely huge and wrong answer, or ends up giving me a integration by parts of something like \int e^(x^2) dx which I can't solve

 

[tiny-tim]

Hi Enzo!

(have an integral: ∫ and try using the X² tag just above the Reply box )

<br /> \int^1_y\int^1_0 x^2*e^{xy} dydx<br />


Answer: 1/2 (e-2)

… ends up … something like \int e^(x^2) dx which I can't solve

Did you change the order of integration first?

I get ∫xe^x2 dx, which is easy

Try again!

 

[HallsofIvy]

Homework Statement

<br /> \int^1_y\int^1_0 x^2*e^{xy} dydx<br />


Answer: 1/2 (e-2)

The Attempt at a Solution

I've tried about 4 ways of doing this, I can't solve it. It either ends up being a completely huge and wrong answer, or ends up giving me a integration by parts of something like \int e^(x^2) dx which I can't solve

This makes no sense. The way you have it written, with the "outer integral" from y to 1, the answer must be a function of y, not a constant. But, as written it does not give "e^{x^2}
\int_{x=y}^1\int_{y= 0}^1 x^2e^{xy}dy dx= \int_{x=y}^1\left(xe^{xy}\right)_0^1 dx
= \int_{x=y}^1 \left(xe^x- x\right)dx
which can be done by a single integration by parts.

If it were
\int_{y=0}^1\int{x=y}^1 x^2e^{xy}dx dy[/itex]<br /> tat can be done by two integrations by parts.