# Find the area bounded by f(x)=sinx and g(x)=cosx from x=pi/4 to x=((5pi)/4.. Make an accurate sketch of the graphs on the axis below?

## Find the area bounded by $f \left(x\right) = \sin x$ and $g \left(x\right) = \cos x$ from $x = \frac{\pi}{4}$ to $x = \left(\frac{5 \pi}{4}\right)$. Make an accurate sketch of the graphs on the axis below.

May 30, 2018

$\textcolor{b l u e}{A = {\int}_{\frac{\pi}{4}}^{5 \frac{\pi}{4}} \sin x - \cos x \cdot \mathrm{dx} = 2 \sqrt{2}}$

#### Explanation:

The Area due to $\text{x-axis}$ between two curves given by:

color(red)[A=int_a^by_2-y_1*dx

The interval of our integral $x \in \left[\frac{\pi}{4} , 5 \frac{\pi}{4}\right]$

now let set up the integral:

$A = {\int}_{\frac{\pi}{4}}^{5 \frac{\pi}{4}} \sin x - \cos x \cdot \mathrm{dx} = {\left[- \cos x - \sin x\right]}_{\frac{\pi}{4}}^{5 \frac{\pi}{4}}$

$\left[\left(- \sin \left(5 \frac{\pi}{4}\right) - \cos \left(5 \frac{\pi}{4}\right)\right) - \left(- \sin \left(\frac{\pi}{4}\right) - \cos \left(\frac{\pi}{4}\right)\right)\right]$

$\left[\frac{2}{\sqrt{2}} + \frac{2}{\sqrt{2}}\right] = \frac{4}{\sqrt{2}} = 2 \sqrt{2}$

$y = \sin x$ blue curve
$y = \cos x$ green curve