find area of region

Question

Find the area of the region bounded by the graphs of y = -x^2 + 3x + 4 and y = 4.

2.7

4.5

28.5

16.5

 

Answer 1

First, set the equations equal to each other.  This will allow us to find the points of intersection.  These intersecting points will serve as the bounds when we integrate the parabolic function.

 

-x² + 3x + 4 = 4

 

 

Make the left side of the equation equal to zero.

 

0 = x² - 3x

 

0 = x(x - 3)

 

x = 0

x = 3

 

 

These x values are your bounds.  So find the definite integral of y=-x²+3x+4  with x=0 as your lower bound and x=3 as your upper bound.  To write this integral in its proper notation, we have

 

 

³₀∫(-x² + 3x + 4)dx