g

f

g g g

f f

Name: ____________________________________________ Date: _________________ Period: __________

Section 7.1: Area of a Region Between Two Curves

With a slight modification, we can change the concept of finding the area of a region under a curve to finding the area of a

region between two curves.

Consider the following graphs of ( )y f x and ( )y g x that are continuous on the interval 2,4 .

4

2

( ) ( )f x g x dx

4

2

( )f x dx

4

2

( )g x dx

Example 1: Finding the Area of a Region Between Curves.

Find the area of the region bounded by the graphs of 2 2, , 0 and 1. y x y x x x

Sketch the graph and shade the region.

x

y

Area of region

between f and g

Area of region

under f = Area of region

under g _

_ =

Area of a Region Between Two Curves If f and g are continuous on [a, b] and ( ) ( )g x f x for all x in [a, b], then the area of the region bounded by the graphs

of f and g and the vertical lines x a and x b is

( ) ( )b

a

A f x g x dx

x

y

x

y

-1 1 2 3

-1

1

2

3

x

y

Example 2: A Region Lying Between Two Intersecting Graphs.

Find the area of the region bounded by the graphs of 2( ) 2 and ( ) . f x x g x x

Sketch the graph and shade the region.

Example 3: A Region Lying Between Two Intersecting Graphs.

Find the area of one of the regions bounded by the graphs of ( ) sin and ( ) cos . f x x g x x

Sketch the graph and shade the region.

-2 -1 1 2

-2

-1

1

2

x

y

x

y

Example 4: Curves That Intersect at More Than Two Points.

Find the area of the regions bounded by the graphs of 3 2 2( ) 3 10 and ( ) 2 . f x x x x g x x x

Sketch the graph and shade the region.

Example 5: Horizontal Representative Rectangles.

Find the area of the region bounded by the graphs of 23 and 1. x y x y

Sketch the graph and shade the region.

x

y

x

y