How Do You Apply the Midpoint Rule for Riemann Integration?

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I am having troubles doing the midpoint rule.. i guess I just need a step by step explanation. for instance

n=3 a=1 b=3 x^2 dx

approximate
 
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Riemann integration for x^2 on the interval (1,3)? Divide your interval into 3 equal sections, determine the middle of each section. Sum the area of the rectangles generated by the height determined by the value of x^2 at the midpoint and the appropriate base.

Really now, this is probably in your textbook.
 
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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