What does the integral represent?

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Manni
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Hey guys I was trying to think conceptually and got a little lost. If R(t) represents the rate of volume per unit time what would it's integral represent?

My initial guess was that it represents the total volume over that interval of time but it feels incomplete, or even wrong.

Help?
 
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Try saying "total volume change" instead of just "total volume". Does it feel better now?
 
Yup, I don't know why I was questioning my definition so much. Thanks Dick :)
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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