# What is the net area between f(x) = cos^2xsinx  and the x-axis over x in [0, 3pi ]?

Sep 21, 2016

$\frac{2}{3}$ areal units.

#### Explanation:

Net area = int f(x) dx, between the limits $0 \mathmr{and} 3 \pi$

$= \int {\cos}^{2} x \sin x \mathrm{dx}$, between the limits $0 \mathmr{and} 3 \pi$

$= - \int {\cos}^{2} x d \left(\cos x\right)$, between the x-limits $0 \mathmr{and} 3 \pi$

$= - \left[{\left(\cos x\right)}^{3}\right]$, between the limits $0 \mathmr{and} 3 \pi$

$= - \left({\left(\cos \left(3 \pi\right)\right)}^{3} - {\left(\cos 0\right)}^{3}\right)$

$= - \left({\left(- 1\right)}^{3} - 1\right)$

$= \frac{2}{3}$

Note that f is periodic with period $2 \pi$ and, Interestingly, the net

periodic area ( here up to $x = 2 \pi$ ) =$\frac{2}{3} - \frac{2}{3} = 0$..