Question

Help Final exam test review question!!
sketch the region
rewrite the integral by switching order
evaluate integral

Answer 1

\[\int\limits_{0}^{1}\int\limits_{y}^{1}x^{2}*e^{xy} dx dy \]

Answer 2

which triangle to use

Answer 3

The bottom triangle because x goes from y to 1 which means FROM the line to the vertical line of x=1.

Answer 4

then i know my integrals change
from 0 to x and from 0 to 1

Answer 5

i dont understand then how to solve the integral

Answer 6

So you have:
\[\int\limits_0^1 \int\limits_0^y x^2 e^{xy} dy dx\]?

Answer 7

First int should be 0 to x ****

Answer 8

yes
i have o to x

Answer 9

i dunno how to solve x^2 e^(xy)
i know its by parts but i am messing up

Answer 10

Well if you integrate y first you don't need by parts:
\[\int\limits_0^x x^2 (e^{x})^y dy = x^2 \ln(e^{x})e^{x y}|_0^x=x^3(e^{x^2}-1)\]
From here you need to be clever:
\[\int\limits_0^1 x^3(e^{x^2}-1)dx; \xi =x^2 \implies d \xi = 2x dx \implies dx = \frac{d \xi}{2x}\]
From this we see the integral transforms into:
\[\frac{1}{2}\int\limits_0^1 \xi(e^{\xi}-1)d \xi\]
From here you can distribute xi and do integrate by parts on the first term and power rule on the second. Sorry it took so long to get back to you.