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

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Algebra 2

Complex conjugate theorem

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Q. Let

f(x)=\frac{1}{2} x^{3}-2 x+3

and

g(x)=x^{2}+\frac{7}{2} x-3

\newline

Find the sum of the areas enclosed by the graphs of

f

and

g

between

x=-3

and

x=4

\newline

Use a graphing calculator and round your answer to three decimal places.

Understand the Problem: Understand the problem. $\newline$ We need to find the sum of the areas enclosed by the graphs of the functions $f(x)$ and $g(x)$ between $x=-3$ and $x=4$ . This involves finding the points of intersection of the two graphs and then integrating the absolute difference of the functions between those points.
Find Intersection Points: Find the points of intersection. $\newline$ To find the points of intersection, we set $f(x)$ equal to $g(x)$ and solve for $x$ . $\newline$ $\frac{1}{2}x^3 - 2x + 3 = x^2 + \frac{7}{2}x - 3$
Rearrange Equation for Roots: Rearrange the equation to find the roots. $\newline$ $(\frac{1}{2})x^3 - x^2 - (\frac{7}{2})x + 6 = 0$ $\newline$ Multiply through by $2$ to clear the fraction: $\newline$ $x^3 - 2x^2 - 7x + 12 = 0$
Use Graphing Calculator: Use a graphing calculator to find the roots. $\newline$ Since the equation is a cubic and may not have a straightforward algebraic solution, we use a graphing calculator to find the roots within the interval $x=-3$ and $x=4$ .
Calculate Areas: Calculate the areas. $\newline$ Once we have the points of intersection, we integrate the absolute difference of the functions between those points to find the enclosed areas. We use the graphing calculator to perform the integration and sum the areas.
Add Individual Areas: Add the areas together. $\newline$ After calculating the individual areas, we add them together to get the total area enclosed by the graphs of $f(x)$ and $g(x)$ between $x=-3$ and $x=4$ .
Round Final Answer: Round the answer to three decimal places. After finding the sum of the areas, we round the answer to three decimal places as instructed.