Hey I've just finished an exam paper and just am stuck with one question. It's something that usually makes sense to me but for some reason I can't get this one:
Let $R = \{(x,y) : x,y ≥ 0, x^2 + y^2 \le 4\} $, which is a circular sector in the first quadrant. Use polar coordinates to evaluate $$\iint_R (x^2 + y^2)e^{\left(x^2+y^2\right)^2} dA.$$
What I've done is $x^2 + y^2 = 4$, so $r = 2$.
since $x, y \ge 0$ then our limits will be $[0, \pi]$
$$\ \int^\pi_0 \int^2_0 r^2e^{r^4}\ dr\ d\theta$$ We substitute $u= r^2, du=2rdr, dr = \frac 1 2 rdu$ to get
$$\int^\pi_0 \int^2_0 ue^{u^2} \frac1 {2\sqrt u}dud\theta $$
Then I'm stuck I think I've gone wrong somewhere with the u's but I'm not sure. any help would be really appreciated!
Hints:
1)since you are in the first quadrant the limits of $\theta$ are $0\le \theta \le \frac{\pi}{2}$.
2) the area element in polar coordinates is $da=rdr d\theta$
so your integral becomes:
$$ \int _0^{\frac{\pi}{2}}\int_0^2 r^3e^{r^4}drd\theta $$
3) use the substitution: $r^4=t$