Start with the y=x² from x=0 to 5 and then spin it around the vertical axis, and you get a bowl shape. It is easier when you want the inside of the bowl than the outside, and that is the case here.
If you look at the bottom of the curve and spin it, you get a circle. This is the key to understanding the integral. The area of a circle is πr². The integral here is π * integral [a,b] (f(y))² dy where "a" and "b" are the terminal y-values. Here a=0 and b=5 and since f(x)=y², f(y) must be x=√y
π (integral)[0,5] (√y)² = π (integral)[0,5] y = π * (y²/2) = (π/2) (5²-0²) = 25π/2
