# Integration by parts: Hydrostatic pressure

The question itself is:

A diving pool that is 4 meters deep, and full of water, has on one of its walls a circular viewing window, tangent to the bottom of the pool, with a radius of .5 meters. Find the force on this window.

I have the integral $$\int_0^1 1000\cdot9.8\cdot(4-y)\sqrt{0.25-y^2}\,\mathrm dx$$, and I have absolutely no idea how to work with it. I've tried multiple different approaches and every time I simply cannot do it. Any help would be appreciated.

• Note that for $y>1/2$, $0.25-y^2<0$. Maybe the upper bound of the integral is not $1$? – Stop hurting Monica Sep 4 at 19:01
• Which approaches have you tried? Can you find $\int 4 \sqrt{0.25 - y^2}\,dy$ and $\int y \sqrt{0.25 - y^2}\,dy$? – Matthew Leingang Sep 4 at 19:08
• @MatthewLeingang I have tried this, but get stuck there as well. I'm severely struggling with this problem on all fronts. – Revellion Sep 4 at 19:10
• “If you can't solve a problem, then there is an easier problem you also can't solve: find it” (Polyà). It sounds like you want to try to find those two antiderivatives. The second is easier than the first. – Matthew Leingang Sep 4 at 19:13
• @Jean-ClaudeArbaut: My certainly led me to reread all of How to Solve It twice and I couldn't find it either way. Then I found this question. Apparently Conway attributed it to Polyà with can't. The way I always thought about it, the idea is simplify the problem as much as you can, without removing the part that makes it difficult. So you can't find $\int\sqrt{0.25-y^2}\,dy$? What about $\int\sqrt{1-y^2}\,dy$ isntead? That's easier, but maintains the essential hardness of the first. – Matthew Leingang Sep 4 at 22:51

$$(1000•9.8•4)-(1000•9.8•y )(0.25-y^2)^{0.5}$$
then use the substitution $$0.25-y^2=y$$ for the second part and integrate the first part manually.
• I multiplied the constants together and then multiplied them by $y$. If you have $a(b-c)$ you can write it as $ab-ac$ – Sina Babaei Zadeh Sep 4 at 20:46
Hint: Split the integrand into two parts: the first a constant multiple of $$\sqrt{0.25-y^2},$$ and the second a constant multiple of $$-y\sqrt{0.25-y^2}.$$ You may do the first by using the substitution $$y=\frac12\sin\phi.$$ To do the second, use the substitution $$u=0.25-y^2.$$