Let f(x)=8-2x and g(x)=x^(3)-7x^(2)+12 x". " Find the sum of the areas enclosed by the graphs of f and g between x=1 and x=4. Use a graphing calculator and round your answer to three decimal places.

Question

Let  f(x)=8-2x and  g(x)=x^(3)-7x^(2)+12 x". " Find the sum of the areas enclosed by the graphs of  f and  g between  x=1 and  x=4. Use a graphing calculator and round your answer to three decimal places.

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Understand the problem: Understand the problem. We need to find the area between the curves of f(x) and g(x) from x=1 to x=4. This is equivalent to integrating the absolute value of the difference between f(x) and g(x) over the interval [1, 4].Set up the integral: Set up the integral to find the area between the curves. The area A between the curves is given by the integral from x=1 to x=4 of the absolute value of the difference between f(x) and g(x), which is |f(x) - g(x)|. So, we have: A = ∫ from 1 to 4 of |f(x) - g(x)| dxCalculate f(x) - g(x): Calculate f(x) - g(x). Before we can integrate, we need to find the expression for f(x) - g(x): f(x) - g(x) = (8 - 2x) - (x^3 - 7x^2 + 12x)              = -x^3 + 7x^2 - 14x + 8Determine points of intersection: Determine the points of intersection between f(x) and g(x). To find the points of intersection, we set f(x) = g(x): 8 - 2x = x^3 - 7x^2 + 12x 0 = x^3 - 7x^2 + 14x - 8 We can use a graphing calculator to find the points of intersection within the interval [1, 4].Use graphing calculator: Use a graphing calculator to find the points of intersection. After plotting the functions on a graphing calculator, we find that the functions intersect at x=1 and x=4. Since these are the limits of our integral, we do not need to split the integral into multiple parts.Integrate to find area: Integrate the function to find the area. We can now integrate the function from x=1 to x=4. Since we are looking for the area, we take the absolute value of the function: A = ∫ from 1 to 4 of |-x^3 + 7x^2 - 14x + 8| dx We can use a graphing calculator to evaluate this integral.Evaluate integral: Evaluate the integral using a graphing calculator. Using the graphing calculator, we find the integral from x=1 to x=4 of |-x^3 + 7x^2 - 14x + 8| dx to be approximately 18.667.

Let $f(x)=8-2 x$ and $g(x)=x^{3}-7 x^{2}+12 x \text {. }$ $\newline$ Find the sum of the areas enclosed by the graphs of $f$ and $g$ between $x=1$ and $x=4$ . $\newline$ Use a graphing calculator and round your answer to three decimal places.

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