Questions: The shaded region between the graphs of y=-2x^2+20 and y=3x-15 is displayed below in orange. The shaded region lies between x= -5 and x= 3.5. Now find the total area of the orange shaded region. Write your answer in exact form. A=

The shaded region between the graphs of y=-2x^2+20 and y=3x-15 is displayed below in orange.

The shaded region lies between x= -5 and x= 3.5.

Now find the total area of the orange shaded region. Write your answer in exact form.
A=

Transcript text: The shaded region between the graphs of $y=-2 x^{2}+20$ and $y=3 x-15$ is displayed below in orange. The shaded region lies between $x=$ $-5$ and $x=$ $3.5$. Now find the total area of the orange shaded region. Write your answer in exact form. $A=$ $\square$

Solution

Solution Steps

Step 1: Find the intersection points

Set the two equations equal to each other and solve for x:

-2x² + 20 = 3x - 15
-2x² - 3x + 35 = 0
2x² + 3x - 35 = 0
(2x - 7)(x + 5) = 0
x = 3.5 or x = -5

Step 2: Set up the integral

The area of the shaded region is given by the integral of the difference between the two functions, from the lower limit of integration (-5) to the upper limit (3.5):

∫[-5, 3.5] (-2x² + 20 - (3x - 15)) dx

Step 3: Evaluate the integral

∫[-5, 3.5] (-2x² - 3x + 35) dx
= [-2x³/3 - (3/2)x² + 35x] [-5, 3.5]
= (-2(3.5)³/3 - (3/2)(3.5)² + 35(3.5)) - (-2(-5)³/3 - (3/2)(-5)² + 35(-5))
= (-85.75 - 18.375 + 122.5) - (83.333 - 37.5 - 175)
= 17.375 - (-129.167)
= 146.542

Final Answer:

A = 146.542 or 729/5

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