009B Sample Final 1, Problem 3

Consider the area bounded by the following two functions:

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle y=\sin x} and $y={\frac {2}{\pi }}x.$

a) Find the three intersection points of the two given functions. (Drawing may be helpful.)

b) Find the area bounded by the two functions.

Foundations:
Recall:
1. You can find the intersection points of two functions, say $f(x),g(x),$
by setting $f(x)=g(x)$ and solving for $x$.
2. The area between two functions, $f(x)$ and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g(x)} , is given by $\int _{a}^{b}f(x)-g(x)~dx$
for $a\leq x\leq b$, where $f(x)$ is the upper function and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle g(x)} is the lower function.

Solution:

(a)

Step 1:
First, we graph these two functions.
|} Step 2:   Setting Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \sin x={\frac {2}{\pi }}x} , we get three solutions: $x=0,{\frac {\pi }{2}},{\frac {-\pi }{2}}.$ So, the three intersection points are $(0,0),{\bigg (}{\frac {\pi }{2}},1{\bigg )},{\bigg (}{\frac {-\pi }{2}},-1{\bigg )}$. You can see these intersection points on the graph shown in Step 1. (b) Step 1:   Using symmetry of the graph, the area bounded by the two functions is given by $2\int _{0}^{\frac {\pi }{2}}{\bigg (}\sin(x)-{\frac {2}{\pi }}x{\bigg )}~dx.$ Step 2:   Lastly, we integrate to get Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{rcl} \displaystyle{2\int_0^{\frac{\pi}{2}}\bigg(\sin (x)-\frac{2}{\pi}x\bigg)~dx} & {=} & \displaystyle{2\bigg(-\cos (x)-\frac{x^2}{\pi}\bigg)\bigg|_0^{\frac{\pi}{2}}}\\ &&\\ & = & \displaystyle{2\bigg(-\cos \bigg(\frac{\pi}{2}\bigg)-\frac{1}{\pi}\bigg(\frac{\pi}{2}\bigg)^2\bigg)}-2(-\cos(0))\\ &&\\ & = & \displaystyle{2\bigg(-\frac{\pi}{4}\bigg)+2}\\ &&\\ & = & \displaystyle{-\frac{\pi}{2}+2}.\\ \end{array}} Final Answer:   (a)  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,0),\bigg(\frac{\pi}{2},1\bigg),\bigg(\frac{-\pi}{2},-1\bigg)} (b)  Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{\pi}{2}+2} Return to Sample Exam