Question

The integral expression to find the area of the region enclosed by y^2=2x and y=x-x^2 using vertical strip

Answer 1

I was trying to color the region, but I failed...
here is the graph, you should be able to identify the requested region.

Answer 2

https://www.desmos.com/calculator/3jmg8uknf1

Answer 3

So, really we are trying to find the region bound by
\(\color{#000000 }{ \displaystyle f(x)=-\sqrt{2x} }\)
\(\color{#000000 }{ \displaystyle g(x)=-x^2+x }\)
and our interval is \(\color{#000000 }{ \displaystyle x\in [0,2] }\)

Notice, on the graph, that g(x) is always above the f(x) for the given interval. So you can just go ahead and do,

\(\color{#000000 }{ \displaystyle \int_0^2 g(x)-f(x)~dx }\)

Answer 4

Go ahead and substitute the functions accordingly, and don't be afraid to get the negative area for this integral because [almost] the entire area is located below the x-axis.