[Math] Please verify the solution: Sketch the region bounded by the curves $y=x^2$ and $y=2x-x^2$ and find the area of the region.

Question

Sketch the region bounded by the curves $y=x^2$ and $y=2x-x^2$ and find the area of the region.

I have an answer but I'd appreciate verification on whether or not I did this right. I've made some stupid mistakes already today so I wouldn't be surprised if this is wrong as well.

Both equations begin at the origin(a) and intersect at a certain value of x(b)

$x^2=2x-x^2$

$2x^2-2x=0$

$2x(x-1)=0$

$x=0,x=1$

So I take the area under the curve from [0,1] of $y=2x-x^2$ and subtracted the area under $y=x^2$

So I got $\int_0^1 (2x-2x^2)dx$

$(x^2-\frac{2}{3}x^3)|_0^1=(1^2-\frac{2}{3}(1)^3)-(0^2-\frac{2}{3}(0)^3)=1-\frac{2}{3}=\frac{1}{3}$

Thus the area bounded by the curves has an area equal to $\frac{1}{3}$

Answer 1

Here's your answer! Hope it helps.