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Let

f(x)=3x^(3)-8x^(2)-3x+12
and

g(x)=4e^((x+1)(x-3))+2x+2.
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 $\newline$ $f(x)=3 x^{3}-8 x^{2}-3 x+12$ $\newline$ and $\newline$ $g(x)=4 e^{(x+1)(x-3)}+2 x+2 .$ $\newline$ Let $R$ and $S$ be the two regions enclosed by the graphs of $f$ and $g$ as shown in the graph. $\newline$ Find the sum of the areas of regions $R$ and $S$ . $\newline$ Use a graphing calculator and round your answer to three decimal places.

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Conjugate root theorems

Full solution

Q. Let

\newline

f(x)=3 x^{3}-8 x^{2}-3 x+12

\newline

and

\newline

g(x)=4 e^{(x+1)(x-3)}+2 x+2 .

\newline

Let

R

and

S

be the two regions enclosed by the graphs of

f

and

g

as shown in the graph.

\newline

Find the sum of the areas of regions

R

and

S

.

\newline

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

Graph Functions: Graph the functions $f(x)$ and $g(x)$ using a graphing calculator. $\newline$ Reasoning: To find the areas of regions $R$ and $S$ , we first need to visualize where these regions are located with respect to the graphs of $f(x)$ and $g(x)$ . $\newline$ Calculation: Use a graphing calculator to plot $f(x) = 3x^3 - 8x^2 - 3x + 12$ and $g(x) = 4e^{(x+1)(x-3)} + 2x + 2$ .
Identify Intersection Points: Identify the points of intersection between $f(x)$ and $g(x)$ . Reasoning: The points of intersection will serve as the limits of integration when calculating the areas of regions $R$ and $S$ . Calculation: Use the graphing calculator's intersection feature to find the $x$ -values where $f(x)$ and $g(x)$ intersect.
Set Up Integral for Region R: Set up the integral to find the area of region R. $\newline$ Reasoning: The area of region R can be found by integrating the difference between $g(x)$ and $f(x)$ over the interval defined by their points of intersection. $\newline$ Calculation: If $x_1$ and $x_2$ are the points of intersection, the area of R is given by the integral from $x_1$ to $x_2$ of $(g(x) - f(x)) \, dx$ .
Set Up Integral for Region S: Set up the integral to find the area of region S. $\newline$ Reasoning: Similarly, the area of region S can be found by integrating the difference between $f(x)$ and $g(x)$ over the interval defined by their points of intersection. $\newline$ Calculation: If $x_2$ and $x_3$ are the points of intersection, the area of S is given by the integral from $x_2$ to $x_3$ of $(f(x) - g(x)) \, dx$ .
Calculate Areas of Regions: Calculate the areas of regions $R$ and $S$ using the integrals. $\newline$ Reasoning: By evaluating the integrals, we can find the exact areas of regions $R$ and $S$ . $\newline$ Calculation: Use the graphing calculator to evaluate the integrals for regions $R$ and $S$ .
Add Total Area: Add the areas of regions $R$ and $S$ to find the total area. $\newline$ Reasoning: The sum of the areas of regions $R$ and $S$ will give us the total area enclosed by the graphs of $f(x)$ and $g(x)$ . $\newline$ Calculation: Add the numerical values obtained from the integrals for regions $R$ and $S$ .

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