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

Net Area $= 3 \pi = 9.42478$

#### Explanation:

Area $= {\int}_{0}^{\pi} x \cdot \sin x$ $\mathrm{dx}$ +${\int}_{\pi}^{2 \pi} x \cdot \sin x$ $\mathrm{dx}$ +${\int}_{2 \pi}^{3 \pi} x \cdot \sin x$ $\mathrm{dx}$

$\int u$ $\mathrm{dv}$ = $u v - \int v$ $\mathrm{du}$

Let $u = x$,

$\mathrm{dv}$ = $\sin x$ $\mathrm{dx}$

$v = - \cos x$

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

$\int x \cdot \sin x$ $\mathrm{dx}$=$- x \cdot \cos x - \int \left(- \cos x\right)$ $\mathrm{dx}$

$= - x \cdot \cos x + \int \cos x$ $\mathrm{dx}$

$= - x \cdot \cos x + \sin x$

Apply the limits $\left[0 , \pi\right] , \mathmr{and} \left[\pi , 2 \pi\right] , \mathmr{and} \left[2 \pi , 3 \pi\right]$

the areas are $\pi \mathmr{and} - 3 \pi \mathmr{and} 5 \pi$

so that the Net Area = $\pi + \left(- 3 \pi\right) + 5 \pi = 3 \pi$