let $f(x)=\frac{1}{2}x^{3}-2x + 3$ and $g(x)=x^{2}+\frac{7}{2}x - 3$. find the sum of the areas enclosed by…

let $f(x)=\frac{1}{2}x^{3}-2x + 3$ and $g(x)=x^{2}+\frac{7}{2}x - 3$. find the sum of the areas enclosed by the graphs of $f$ and $g$ between $x=-3$ and $x = 4$. use a graphing calculator and round your answer to three decimal places.

let $f(x)=\frac{1}{2}x^{3}-2x + 3$ and $g(x)=x^{2}+\frac{7}{2}x - 3$. find the sum of the areas enclosed by the graphs of $f$ and $g$ between $x=-3$ and $x = 4$. use a graphing calculator and round your answer to three decimal places.

Answer

Explanation:

Step1: Find the difference function

Let $h(x)=f(x)-g(x)=\frac{1}{2}x^{3}-2x + 3-(x^{2}+\frac{7}{2}x - 3)=\frac{1}{2}x^{3}-x^{2}-\frac{11}{2}x + 6$.

Step2: Use definite - integral to find the area

The area $A=\int_{-3}^{4}|h(x)|dx$. We can use a graphing calculator to evaluate $\int_{-3}^{4}\left|\frac{1}{2}x^{3}-x^{2}-\frac{11}{2}x + 6\right|dx$. Using a graphing calculator (such as TI - 84 Plus: enter the function $y = \left|\frac{1}{2}x^{3}-x^{2}-\frac{11}{2}x + 6\right|$ and then use the integral function $\int_{a}^{b}y\mathrm{d}x$ with $a=-3$ and $b = 4$), we get the result.

Answer:

$71.583$