1. Chapter 19
Area of a Region
Between Two Curves

2. - - = f f f g g g Area of region between f and g Area of region
under f(x) =
Area of region
under g(x)

3. Ex. Find the area of the region bounded by the graphs
of f(x) = x2 + 2, g(x) = -x, x = 0, and x = 1 .
Area = Top curve – bottom curve
f(x) = x2 + 2
g(x) = -x

4. Find the area of the region bounded by the graphs of
f(x) = 2 – x2 and g(x) = x
First, set f(x) = g(x) to
find their points of
intersection.
2 – x2 = x
0 = x2 + x - 2
0 = (x + 2)(x – 1)
x = -2 and x = 1
Use technology

5. Find the area of the region between the graphs of
f(x) = 3x3 – x2 – 10x and g(x) = -x2 + 2x
Again, set f(x) = g(x)
to find their points of
intersection.
3x3 – x2 – 10x = -x2 + 2x
3x3 – 12x = 0
3x(x2 – 4) = 0
x = 0 , -2 , 2
Note that the two graphs switch at the origin.

6. Now, set up the two integrals and solve.

8. Find the area of the region that is bounded by the graphs of
First, look at the graph
of these two functions.
Determine where they
intersect.
(endpoints not given)

9. Second, find the points of intersection by setting f (x) = g (x) and solving.

10. Lastly, compute the integral
Lastly, compute the integral. Note that on [0, 2], f (x) is the upper graph. ( 2 x + 1 ) - é ë ù û ò d = 3 ê ú æ è ç ö ø ÷ 4 8

11. Homework:
Page 481
Evens
Due Tuesday