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

[Enzo]

Homework Statement

$$\int^1_y\int^1_0 x^2*e^{xy} dydx$$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 )

$$\int^1_y\int^1_0 x^2*e^{xy} dydx$$


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

$$\int^1_y\int^1_0 x^2*e^{xy} dydx$$


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