A Community Problem Contributed By Our Member debroy0611

image containing math problem statement
The easiest problem you have ever seen, if you catch the trick, a very difficult one otherwise.
Problem contributed by debroy0611
We will solve this problem using a simple geometric approach, instead of attempting a calculus based approach:

Let:
$y = \sqrt{x(1-x)}$

$\Rightarrow y^2 = x - x^2$

$\Rightarrow x^2 - x  + y^2 = 0$

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$\Rightarrow x^2 - 2 \cdot \dfrac{1}{2} \cdot x + \dfrac{1}{4} + y^2 = \dfrac{1}{4}$

$\Rightarrow \left(x - \dfrac{1}{2} \right)^2 + (y - 0)^2 = \left(\dfrac{1}{2}\right)^2$

The above is the equation of a circle with radius $\dfrac{1}{2}$, and centered at $\left( \dfrac{1}{2}, 0 \right)$

Since the problem is asking for the integral of $\sqrt{x(1-x)}$, this is given by the positive half of the circle.

Therefore, the area is:
$\dfrac{1}{2} \pi \left( \dfrac{1}{2} \right)^2$

$= \dfrac{\pi}{8} \approx 0.39$