# What is the net area between f(x)=xsinx in x in[0,4pi]  and the x-axis?

Jun 26, 2016

$= - 4 \pi$

#### Explanation:

graph{x sin x [-36.5, 36.48, -18.24, 18.26]}

you want the net area so you don't mind that the positive and negative values net off. you need to know, therefore

${\int}_{0}^{4 \pi} x \sin x \mathrm{dx}$

we can do this by IBP using the idea that $\int u v ' \mathrm{dx} = u v - \int u ' v \mathrm{dx}$

$u = x , u ' = 1$
$v ' = \sin x , v = - \cos x$

${\int}_{0}^{4 \pi} x \sin x \mathrm{dx}$
$= {\left[- x \cos x\right]}_{0}^{4 \pi} - {\int}_{0}^{4 \pi} - \cos x \mathrm{dx}$

$= {\left[- x \cos x\right]}_{0}^{4 \pi} + {\int}_{0}^{4 \pi} \cos x \mathrm{dx}$

$= {\left[- x \cos x + \sin x\right]}_{0}^{4 \pi}$

so we evaluate

${\left[- x \cos x + \sin x\right]}_{0}^{4 \pi}$

$= - 4 \pi$