$$\int\limits_{}^{}\int\limits_{D}^{}y ^{2}e ^{xy}dA$$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.

Question

$$\int\limits_{}^{}\int\limits_{D}^{}y ^{2}e ^{xy}dA$$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.

anonymous · Answer

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.

turingtest · Answer

|dw:1352586388142:dw|

turingtest · Answer

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

turingtest · Answer

does that make any sense to you?

anonymous · Answer

Yep, I get that.

turingtest · Answer

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

anonymous · Answer

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

turingtest · Answer

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

turingtest · Answer

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

anonymous · Answer

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

turingtest · Answer

yes

anonymous · Answer

Thank you!

turingtest · Answer

welcome :D