Integral Is Area

I learned of this example from Kunihiko Chikaya's post on facebook. The task is to compute the integral

$\displaystyle\int_{a}^{b}\sqrt{(x-a)(b-x)}dx.$

The solution is to recollect the geometric meaning of the definite integral.

Compute the followng integral

$\displaystyle\int_{a}^{b}\sqrt{(x-a)(b-x)}dx.$

Solution

Let $y=\sqrt{(x-a)(b-x)}.$ For $y\gt 0,\;$ this is equivalent to $\displaystyle\left(x-\frac{a+b}{2}\right)^2+y^2=\left(\frac{b-a}{2}\right)^2\;$ which is a semicircle with center at $\displaystyle (\frac{a+b}{2},0)\;$ and radius $\displaystyle\frac{b-a}{2}:$

integral is area

Thus, the integral is half the area of a circle with radius $\displaystyle\frac{b-a}{2}:$

$\displaystyle\frac{1}{2}\pi\left(\frac{b-a}{2}\right)^2=\frac{\pi (b-a)^2}{8}.$

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