Calculating Volume of a Sphere Cap: What Am I Missing?

[GunnaSix]

Homework Statement

For a sphere of radius r find the volume of the cap of height h.

Homework Equations

The Attempt at a Solution

I can get it down to $$V \ = \ \pi \int_{r-h}^r (\pi r^2 - \pi y^2)dy \ = \ \pi {\left[(r^2y-\dfrac{1}{3} y^3) \right] }_{r-h}^r$$

I expanded this to $$V=\pi (-\dfrac{2}{3} r^3+2r^2h-rh^2-\dfrac{1}{3}h^3)$$

but the book has $$V=\pi h^2(r-\dfrac{1}{3} h)$$

What am I missing/doing wrong?

 

[Dick]

The calculus part is ok, except that you mistyped your integrand. The problem is in the algebra leading to your expression for V. For example, I can tell by looking at your integral expression that there shouldn't be any r^3 term in the result.

 

[payumooli]

do the expansion again?

 

[GunnaSix]

Yeah I got it. Forgot to carry the subtraction in the $$(r-h)^3$$ expansion twice in a row and just assumed I was missing something instead of checking my algebra again. I should have seen that the $$r^3$$ expressions would cancel out. Sorry.