# Choosing the order of integration with double integration

1. Jan 6, 2010

### 8614smith

1. The problem statement, all variables and given/known data
Sketch the regions of integration, and evaluate the integral by choosing the best order of integration.
$$\int^{2\sqrt{ln2}}_{0}\int^{\sqrt{ln2}}_{y/2}exp(x^2)dxdy$$

2. Relevant equations
integration by parts

3. The attempt at a solution
ive changed the order of integration and done the inner integral with respect to y to get this far..
$$\int^{\sqrt{ln2}}_{y/2}\int^{2\sqrt{ln2}}_{0}exp(x^2)dydx=\sqrt{ln2}\int^{2\sqrt{ln2}}_{y/2}exp(x^2)dx$$

and now when i do a u substitution
$$u=x^2$$
$$du=2xdx$$
and that is how far i can get as i can't put the 'du' into the equation as it has a '2x' in it and i will be integrating with respect to 'u'. Integrating it the original way round just gets me to this problem straight away.

I just can't see any way of getting rid of the 2x?!

2. Jan 6, 2010

### rock.freak667

If you change the order by which you integrate, the limits will change. You need to find the new limits.

Also ∫ex2dx does not exist in terms of elementary functions.

3. Jan 6, 2010

### Staff: Mentor

Make sure you sketch the region of integration, as step that many students skip, believing it to be unimportant.

4. Jan 6, 2010

### 8614smith

that seems so obvious now but you wouldn't believe how long ive been looking at it, the 'cancelling 2x' comes from when you change the limits. thanks guys!