## anonymous 3 years ago $\int\limits_{}^{}\int\limits_{D}^{}y ^{2}e ^{xy}dA$D is bounded by y=x. y=4, x=0 Set up iterated integrals for both orders of integration. Then evaluate the integral using the easier order and explain why it is easier.

1. anonymous

I know I can do... $\int\limits_{0}^{4}\int\limits_{x}^{4}y ^{2}e ^{xy}dydx$but I'm not sure what they mean by both orders of integration and how to find the other orders.

2. TuringTest

|dw:1352586388142:dw|

3. TuringTest

$x\le y\le4,~~0\le x\le4\implies0\le x\le y,~~0\le y\le4$

4. TuringTest

does that make any sense to you?

5. anonymous

Yep, I get that.

6. TuringTest

so if you switch the order, those would be your bounds where are you stuck?

7. anonymous

How would I break up the integral into iterated sections since the x is in e^(xy)?

8. TuringTest

$\int_0^4\int_0^yy^2e^{yx}dxdy=\int_0^4y\left[\int_0^y ye^{yx}dx\right]dy$

9. TuringTest

notice that$\frac{\partial}{\partial x}e^{yx}=ye^{yx}$

10. anonymous

Alright, so I would have.. $\int\limits_{0}^{4}ydy*[e ^{xy}|0 \ \to\ y]?$

11. TuringTest

yes

12. anonymous

Thank you!

13. TuringTest

welcome :D