The area enclosed by the curve y^2 = x(1 − x) is given by

Question

asked Dec 3, 2019 141k views

1 Answer

Vasion · Answer 1 · 2019-12-09T08:45:42+0000

y^2 = x(1-x) = x- x^2

x^2 + y^2 = x

That's a circle, as we can see by the usual completing of the square,

x^2 - x + y^2 = 0

$x^2 - x + \frac 1 4 + y^2 = \frac 1 4$

$(x - \frac 1 2)^2 + y^2 = (\frac 1 2 )^2$

That's the circle of radius
$\frac 1 2$ centered at
$(\frac 1 2, 0)$

It's a pretty good substitute for the unit circle with some interesting trigonometry of its own.

Anyway its radius is a half so its area is
$\pi r^2 = \pi (\frac 1 2 )^2 = (\pi)/(4)$

Answer:
$\quad \pi / 4$