Question

Consider the function $f\left(x\right)=x\cos 2x$.

(a) Find the signed area between the graph of $f\left(x\right)$and the $x$-axis on $\left[0,\frac{3\mathrm{\pi }}{4}\right]$shown in the figure next at the left.

(b) Find the area between the graph of$f\left(x\right)$and the graph ofrole="math" localid="1650063304032" $g\left(x\right)=-\cos 2x$on$\left[0,\frac{3\mathrm{\pi }}{4}\right]$shown in the figure next at the right.

Short answer

Part (a). The required area is $-1.4275$.

Part (b). The required area is$-1.9275$.

Step-by-step solution

Part (a) Step 1. Given information.

The given function is$f\left(x\right)=x\cos 2x$and$g\left(x\right)=-\cos 2x$.

Part (a) Step 2. Find the area.

The required area is:

$\begin{array}{rcl}{\int }_0^{\frac{3\mathrm{\pi }}{4}}\left(x\cos 2x\right)dx& =& {\left[x\int \cos 2xdx-\int \left\{\frac{d}{dx}\left(x\right)\int \cos 2xdx\right\}dx\right]}_0^{\frac{3\mathrm{\pi }}{4}}\\ & =& {\left[x·\frac{\sin 2x}{2}-\int \frac{\sin 2x}{2}dx\right]}_0^{\frac{3\mathrm{\pi }}{4}}\\ & =& {\left[x·\frac{sin2x}{2}+\frac{1}{4}\cos 2x\right]}_0^{\frac{3\mathrm{\pi }}{4}}\\ & =& \left[\frac{3\mathrm{\pi }}{4}·\frac{sin\left({\displaystyle \frac{3\mathrm{\pi }}{2}}\right)}{2}+\frac{1}{4}\cos \left(\frac{3\mathrm{\pi }}{2}\right)-0-\frac{1}{4}\cos \left(0\right)\right]\\ & =& \frac{1}{2}·\frac{3\mathrm{\pi }}{4}(-1)-\frac{1}{4}\\ & =& -\frac{3\mathrm{\pi }}{8}-\frac{1}{4}\\ & =& -1.4275\end{array}$

Hence, the required area is$A_1=-1.4275$.

Part (b) Step 1. Given information.

The given function is$f\left(x\right)=x\cos 2x$and$g\left(x\right)=-\cos 2x$.

Part (b) Step 2. First, find the definite integral ∫03π4-cos2x dx.

$\begin{array}{rcl}{A}_2& =& {\int }_0^{\frac{3\mathrm{\pi }}{4}}-\cos 2xdx\\ & =& {\left[\frac{-\sin 2x}{2}\right]}_0^{\frac{3\mathrm{\pi }}{4}}\\ & =& \left[\frac{-\sin \left({\displaystyle \frac{3\mathrm{\pi }}{2}}\right)}{2}-0\right]\\ & =& \frac{1}{2}\\ & =& 0.5\end{array}$

Part (b) Step 3. Find the area between the graph of f(x) and g(x).

$\begin{array}{rcl}A& =& {\int }_0^{\frac{3\mathrm{\pi }}{4}}f\left(x\right)-{\int }_0^{\frac{3\mathrm{\pi }}{4}}g\left(x\right)\\ & =& A_1-A_2\\ & =& -1.4275-0.5\\ & =& -1.9275\end{array}$