using integration to find pressure at depth?

Question

So, I'm doing a physics problem, and I ran into a bit of a problem. I am trying to find the force water exerts on a dam - I found that the total force is equal to
(where H = height of water, W = width)

F = (water density)( gravity )( W ) times the

integral from 0 to H of (H - y) dy. where y is the vertical axis

How would I find that integral?

thanks!

Answer 1

I'm assuming you are not a calculus student (otherwise, you really should know this already, because it's very basic calc)

integral(H-y)dy
= integra(H)dy - integral(y)dy
= Hy – y²/2

Applying the limits (0,H)

integral = (H*H – H²/2) – (0–0)
= H²/2

Answer 2

integeral [0..H] of (H-y) dy =
integeral [0..H] of H dy - integeral [0..H] of ydy =
Hy -1/2y² [from 0 to H] =
[H² - 1/2H²] - [0 - 1/2 * 0] =
H²/2