Determine the area bounded by the curves $y=2x$ and $y=x^2$ and the parabaloid $z=x^2+y^2$

Question

Determine the area bounded by the curves $y=2x$ and $y=x^2$ and the parabaloid $z=x^2+y^2$

I'm not really sure what my function that I'm integrating is supposed to be I believe I want to integrate on the set $S=\{(x,y): 0\leq x\leq 2, 0\leq y\leq 4\}$?

and evalute $\int\int_S x^2+y^2 dA$?

In which case I evaluated this to be $8\frac{20}{3}$.

Answer 1

If $2x=x^2 $ then $x=0,2$. We know that : $2x>x^2$ for $x\in[0,2]$ so :

$$Area=\int_{0}^{2}\int_{x^2}^{2x}x^2+y^2dydx=\int_{0}^{2}\frac{14}{3}x^3-x^4-\frac{1}{3}x^6dx=\frac{56}{3}-\frac{32}{5}-\frac{128}{21}$$