I= ∫∫R(x+y)dAwhereRis the region bounded by the curvey=x² and the liney= 1.

(a) CalculateIby integrating first inyand then inx.

(b) CalculateIby integrating first inxand then iny.

This is what I got for (a): Limits for y arex² =< y =< 1, making this a parabola from origin to y=1. solvingx²=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 solvingx² = 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?