Proving Integrability: f on [a,b] with c ∈ [a,b]

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Homework Statement


Prove if f is integrable on [a,b] then \exists c a\leq c \leq b so that \int_a^cf = \int_c^bf

The Attempt at a Solution


I know that you have to use the first fundamental theorem of calculus and start with:
g(x) = \int_a^xf(t)dt-\int_x^bf(t)dt

I'm not sure how to get started and to use this fact to help prove it. Any help would be really appreciated. Thanks!
 
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g(x) is a continuous function, right? Use the intermediate value theorem. What are g(a) and g(b)?
 

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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