# What is the net area between f(x) = cos2x-sinx  and the x-axis over x in [0, 3pi ]?

Sep 18, 2017

$f \left(x\right) = \cos 2 x - \sin x$
area between $f \left(x\right)$ and X-axis is
$y = {\int}_{0}^{3 \pi} \left(\cos 2 x - \sin x\right) \mathrm{dx}$
$y = {\left[\frac{\sin 2 x}{2} - \left(- \cos x\right)\right]}_{0}^{3 \pi}$
$y = {\left[\frac{\sin 2 x}{2} + \cos x\right]}_{0}^{3 \pi}$
$y = \left[\frac{\sin 2 \left(3 \pi\right)}{2} + \cos \left(3 \pi\right)\right] - \left[\frac{\sin 2 \left(0\right)}{2} + \cos \left(0\right)\right]$
$y = \left[\frac{\sin 6 \pi}{2} + \cos \left(3 \pi\right)\right] - \left[1\right]$
$y = \left[\frac{0}{2} + \cos \left(2 \pi + \pi\right)\right] - \left[1\right]$
y=[(cos(pi)]-[1]
$y = \left[- 1\right] - \left[1\right]$
$y = - 2$

Area is Positive
Area$= 2 u n i {t}^{2}$