Double integration of an exp function

[Vandella]

Homework Statement

Please help I need to find the area of e^(-x^2) with the domain 0 is less than or equal to y which is less than or equal to 1 and y is less than or equal to x which is less than or equal to one

Homework Equations

The Attempt at a Solution

I set up the equation ∫ ∫ e^(-x^2) dydx with values y=x and y=0 and x=1 and x=0
attempted to integrate but struggling.

 

[genericusrnme]

Let's see your attempt at integration then

 

[Vandella]

Well I started by integrating with respect to y and got ye^(-x^2)dx,I then solved for the values y=x and y=0 leaving xe^(-x^2)dx and unsure of if my first steps are correct and how to continue.
My idea for a next step would be integration by substitution u=-x^2 but unsure as to whether that would work

 

[genericusrnme]

why -x^2?
I'd just go for x^2

Try it and see what happens

 

[Vandella]

Ok here goes u= x^2 that gives ∫ e^(-u) integration by substitution gives me -(e^x^2)/2
Possibly :)

Now do I just solve for x=1 and x=y?

 

[genericusrnme]

When you did your first integration, you essentially integrated the strip from 0 to y=x, in the second integration you are taking all of those strips from 0 to 1, you don't need to solve for x=1 or x=y, you just need to evaluate the integral from 0 to 1

 

[Vandella]

Sorry when I said solve I meant evaluate.
Could I switch the order of integration so when I evaluate after integrating a second time I lose x and y from the equation?

 

[genericusrnme]

Try it and see what happens
You'll have trouble evaluating it if you do the dx integration first however, the integral of e^(-x^2)dx isn't nice, it isn't solvable in terms of elementary functions. If you're interested, the name of the solution is the error function.

 

[Vandella]

Thanks for your help will just leave it now and not confuse myself even more

 

[genericusrnme]

Thanks for your help will just leave it now and not confuse myself even more

The best way to get use to it is to just play about with it and see what you end up with.