Integration is used to find the area between a curve and the x- or y-axis, and the shaded region here is exactly that area (between the curve and the x-axis, specifically). So all we have to do is integrate 4x-x^2.
We also need to figure out the bounds of integration. From your diagram, I see that the bounds are the zeros of the function 4x-x^2; however, we have to find out numerical values for these zeros, which we can accomplish by factoring 4x-x^2 and setting it equal to zero:
4x-x^2=0
x(4-x)=0
x=0color(white)(XX)andcolor(white)(XX)x=4
We will therefore integrate 4x-x^2 from 0 to 4:
int_0^4 4x-x^2dx
=[2x^2-x^3/3]_0^4-> using reverse power rule (intx^ndx=(x^(n+1))/(n+1))
=((2(4)^2-(4)^3/3)-(2(0)^2-(0)^3/3))
=((32-64/3)-(0))
=32/3~~10.67
