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

Dec 5, 2015

$2$ units

#### Explanation:

We may rewrite this function by making use of the double angle trig identities :

$f \left(x\right) = \sin x \cos x = \frac{1}{2} \sin \left(2 x\right)$

We now draw the graph of this function to help decide limits of integration

graph{1/2sin(2x) [-4.933, 4.934, -2.466, 2.467]}

Since this graph is symmetric with period $\pi$, the graph makes 2 full cycles in the interval $\left[0 , 2 \pi\right]$

The net are area bounded between the graph and the x-axis is hence 4 times the area bounded in $\left[0 , \frac{\pi}{2}\right]$.

ie. $A r e a = 4 {\int}_{0}^{\frac{\pi}{2}} \frac{1}{2} \sin \left(2 x\right) \mathrm{dx}$

$= - 4 \cdot \frac{1}{2} \cdot \frac{1}{2} {\left[\cos 2 x\right]}_{0}^{\frac{\pi}{2}}$

$= - \left(- 1 - 1\right)$

$= 2$