Reversing Order of Integration: Evaluating an Intgral

[shards5]

Homework Statement

Evaluate the integral by reversing the order of integration.
$$\int^{3}_{0}\int^{9}_{y^2} y cos(x^2) dydx$$

Homework Equations

...?

The Attempt at a Solution

Drawing the picture out we get a sideways parabola.
From the picture I get the following intervals of integration.
0 $$\leq$$ y $$\leq$$ $$\sqrt{x}$$
0 $$\leq$$ x $$\leq$$ 9
Using the above I get the following integral.
$$\int^{9}_{0}\int^{sqrt(x)}_{0} y cos(x^2) dxdy$$
After the first integration I get.
$$\frac{y^2}{2}$$ cos(x^2)
Plugging in $$\sqrt{x}$$ and 0 I get the following resulting integral.
$$\int^{9}_{0} x/2 * cos(x^2) dy$$
And here is my problem. It has been a while since I took my calculus II so I don't remember how to integrate the above and I am also not sure if I set my intervals of integration correctly.

 

[SVXX]

Wait a minute...the second last integral is first done wrt y and then the last integral is done wrt x. How did you get cos(x²)? cos(x) would be a constant wrt y. After correction, the last integral will simply become a by parts integral.

 

[shards5]

It was cos(x^2), and I think I got it using integration by substitution. Thanks a lot.