let (f(x)=sin(pisqrt{\frac{x}{2}})) and (g(x)=-\frac{4}{21}cdot x(x - \frac{10}{3})). let (r) and (s) be the…

let (f(x)=sin(pisqrt{\frac{x}{2}})) and (g(x)=-\frac{4}{21}cdot x(x - \frac{10}{3})). let (r) and (s) be the two regions enclosed by the graphs of (f) and (g) as shown in the graph. find the sum of the areas of regions (r) and (s). use a graphing calculator and round your answer to three decimal places.

let (f(x)=sin(pisqrt{\frac{x}{2}})) and (g(x)=-\frac{4}{21}cdot x(x - \frac{10}{3})). let (r) and (s) be the two regions enclosed by the graphs of (f) and (g) as shown in the graph. find the sum of the areas of regions (r) and (s). use a graphing calculator and round your answer to three decimal places.

Answer

Explanation:

Step1: Recall area - between - curves formula

The area between two curves $y = f(x)$ and $y = g(x)$ from $x=a$ to $x = b$ is given by $A=\int_{a}^{b}|f(x)-g(x)|dx$. To find the sum of the areas of regions $R$ and $S$, we need to integrate $|f(x)-g(x)|$ over the interval from $x = 0$ to $x=4.5$.

Step2: Set up the integral

We have $A=\int_{0}^{4.5}\left|\sin\left(\pi\sqrt{\frac{x}{2}}\right)-\left(-\frac{4}{21}x\left(x - \frac{10}{3}\right)\right)\right|dx=\int_{0}^{4.5}\left|\sin\left(\pi\sqrt{\frac{x}{2}}\right)+\frac{4}{21}x\left(x - \frac{10}{3}\right)\right|dx$. First, expand $\frac{4}{21}x\left(x-\frac{10}{3}\right)=\frac{4}{21}x^{2}-\frac{40}{63}x$.

Step3: Use a graphing - calculator

Using a graphing calculator (such as TI - 84 Plus: enter $Y_1=\sin(\pi\sqrt{\frac{X}{2}})$ and $Y_2 =-\frac{4}{21}X*(X - \frac{10}{3})$, then use the integral function $\int_{0}^{4.5}|Y_1 - Y_2|dX$). The value of the integral $\int_{0}^{4.5}\left|\sin\left(\pi\sqrt{\frac{x}{2}}\right)+\frac{4}{21}x^{2}-\frac{40}{63}x\right|dx\approx 3.133$.

Answer:

$3.133$