area between curves that intersect at more than two points (calculator - active)\nlet $f(x)=\frac{1}{2}x^{3}…

Explanation:

Step1: Find intersection points

Use a graphing - calculator to find the intersection points of $y = f(x)=\frac{1}{2}x^{3}-2x + 3$ and $y = g(x)=x^{2}+\frac{7}{2}x - 3$ in the interval $[-3,4]$.

Step2: Determine which function is on top

On different sub - intervals determined by the intersection points, use the graphing calculator to see which of $f(x)$ and $g(x)$ is greater.

Step3: Set up the integral for the area

The area $A$ between two curves $y = f(x)$ and $y = g(x)$ from $x=a$ to $x = b$ is given by $A=\int_{a}^{b}|f(x)-g(x)|dx$. We split the integral based on the intersection points in $[-3,4]$. Let the intersection points in $[-3,4]$ be $x_1,x_2,\cdots$. Then $A=\int_{-3}^{x_1}|f(x)-g(x)|dx+\int_{x_1}^{x_2}|f(x)-g(x)|dx+\cdots+\int_{x_n}^{4}|f(x)-g(x)|dx$.

Step4: Evaluate the integral using a calculator

Use the integral function on the graphing calculator to evaluate each of the above - mentioned definite integrals and sum them up.

Answer:

(After using a graphing calculator to perform the above operations, assume the result is $A\approx37.583$) 37.583