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

Jun 7, 2017

$A = \frac{1}{3} {\sin}^{3} 3 \pi$

#### Explanation:

The question is asking us to find ${\int}_{0}^{3 \pi} \cos x {\sin}^{2} x$ $\mathrm{dx}$

$\int \cos x {\sin}^{2} x$ $\mathrm{dx}$

Let $u = \sin x$

$\mathrm{du} = \cos x$ $\mathrm{dx}$

$\mathrm{dx} = \sec x$ $\mathrm{du}$

$\int \cos x {\sin}^{2} x$ $\mathrm{dx} = \int {u}^{2} \cos x \times \sec x$ $\mathrm{du}$
$= \int {u}^{2}$ $\mathrm{du} = \frac{1}{3} {u}^{3} + \text{c" = 1/3sin^3x +"c}$

${\int}_{0}^{3 \pi} \cos x {\sin}^{2} x$ $\mathrm{dx} = {\left[\frac{1}{3} {\sin}^{3} x\right]}_{0}^{3 \pi}$
$= \frac{1}{3} {\sin}^{3} 3 \pi - 0 = \frac{1}{3} {\sin}^{3} 3 \pi$