
double integral
I = ∫∫R (x + y)dA where R is the region bounded by the curve y = x² and the line y = 1.
(a) Calculate I by integrating first in y and then in x.
(b) Calculate I by integrating first in x and then in y.
This is what I got for (a): Limits for y are x² =< y =< 1, making this a parabola from origin to y=1. solving x²=1 (upper limit of y) gives the limits for x 1=<x=<1.
$\displaystyle \int_{1}^{1}\int_{x^2}^{1} (x+y) $
$\displaystyle \int_{1}^{1} (xy+ \frac{y^2}{2})\bigg_{x^2}^{1} $
$\displaystyle \frac{1}{2} \int_{1}^{1} (x^4  2x^3 + 2x +1)$
$\displaystyle \frac{1}{2} (\frac{x^5}{5}  \frac{x^4}{2} + x^2 +x)\bigg_{1}^{1}$
$\displaystyle =\frac{4}{5} $
which I hope is correct!
I'm stuck on (b)
What should I make the limits of x & y to?
are the limits for x: $\displaystyle 1 \leq x \leq \sqrt{y}$? This is from solving x² = y, making $\displaystyle x=\sqrt{y}$
But what do I make the limits for y? If I use $\displaystyle 0 \leq y \leq 1$ I get a different answer from (a). I get 13/20.
where am I going wrong?

$\displaystyle x^2=y ~\implies~ x = \sqrt{y} ~\implies~ x = \pm \sqrt{y}$
So, this means that $\displaystyle x=\sqrt{y}$ where x is positive $\displaystyle x=\sqrt{y}$ where x is negative.
Therefore, the endpoints will be
$\displaystyle \int_0^1 \int_{\sqrt{y}}^{\sqrt{y}} (x+y) \, dx \, dy$

D'oh! It's so obvious. Thanks heaps for that. It was bugging me no end I've got this assignment due in a couple of days.