Double Intergral with Substitution

by BenMcC
Tags: integral calculus, multivar calculus, substitution
BenMcC is offline
Nov20-12, 10:09 AM
P: 8
I have a problem with Double Integral that I can not seem to get correct.

4 2
∫ ∫ e^(y^2)dydx
0 (x/2)

The answer is (e^4)-1, but I can't seem to get the Substitution at all right. I have literally spent hours on this problem. Any help would be greatly appreciated, its also due by 3:30, so I am kind of limited by time. Thanks
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mfb is offline
Nov20-12, 10:26 AM
P: 10,813
3:30 in which time zone? ;)
Here at me, you asked your question at 5:09 PM.

Converted in a format that is easier to read:
$$\int_0^4 \left(\int_{x/2}^2 e^{y^2} dy\right)dx$$
Did you draw a sketch of the integration area (in the x,y-plane)? The integral is equivalent to
$$\int_0^2 \left(\int_{0}^{2y} e^{y^2} dx\right)dy$$
There, both integrals are easy to evaluate in that order.
BenMcC is offline
Nov20-12, 10:31 AM
P: 8
I have 4 hours, so I have time. I'm just really confused how to do the initial integral. I tried u substitution of u*dv=uv-∫v*du, and I can't get it to come out quite right

mfb is offline
Nov20-12, 11:36 AM
P: 10,813

Double Intergral with Substitution

You cannot find an antiderivative of ##e^{y^2}##. It is nice to try, but you won't get a result unless you just define it (but that does not help here).
The change of the integration order (or something equivalent) is the key point here. It allows to evaluate one integral and simplifies the other one.
BenMcC is offline
Nov20-12, 11:52 AM
P: 8
It's possible. It says to change the order of integration, and I have no idea how to set that up
Vorde is offline
Nov20-12, 05:04 PM
Vorde's Avatar
P: 784
As you may or may not see, evaluating it with regards to x first will let you solve the whole thing. The trick is just the figure out the new bounds on the integrals - it's not as simple as just switching the order.

What I would do is the draw a graph and draw the lines x=4, x=0, y=2 and y=x/2.
Then think about what the bounds would be if you were looking at it 'sideways'.

EDIT: Just realized mfb gave you the integral already, so you don't need what I said.

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