area between curves that intersect at more than two points (calculator - active)\nlet f(x)=sin(π√(x/2)) and…

area between curves that intersect at more than two points (calculator - active)\nlet f(x)=sin(π√(x/2)) and g(x)= - 4/21·x(x - 10/3). let r and s be the two regions enclosed by the graphs of f and g as shown in the graph.\nfind the sum of the areas of regions r and s.\nuse a graphing calculator and round your answer to three decimal places.

area between curves that intersect at more than two points (calculator - active)\nlet f(x)=sin(π√(x/2)) and g(x)= - 4/21·x(x - 10/3). let r and s be the two regions enclosed by the graphs of f and g as shown in the graph.\nfind the sum of the areas of regions r and s.\nuse a graphing calculator and round your answer to three decimal places.

Answer

Explanation:

Step1: Recall area - between - curves formula

The area $A$ between two curves $y = f(x)$ and $y = g(x)$ from $x=a$ to $x = b$ is given by $A=\int_{a}^{b}|f(x)-g(x)|dx$. The intersection points of $f(x)=\sin(\pi\sqrt{\frac{x}{2}})$ and $g(x)=-\frac{4}{21}x(x - \frac{10}{3})$ are $x = 0$ and $x=4.5$.

Step2: Set up the integral

The sum of the areas of regions $R$ and $S$ is $A=\int_{0}^{4.5}|\sin(\pi\sqrt{\frac{x}{2}})-(-\frac{4}{21}x(x - \frac{10}{3}))|dx=\int_{0}^{4.5}|\sin(\pi\sqrt{\frac{x}{2}})+\frac{4}{21}x(x - \frac{10}{3})|dx$.

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 $\int_{0}^{4.5}|Y_1 - Y_2|dX$).

Answer:

$3.177$