1. Section 7.2a
Area between curves

2. Area Between Curves
Suppose we want to know the area of a region that is bounded
above by one curve, y = f(x), and below by another, y = g(x):
Upper curve
First, we partition the region
into vertical strips of equal
width and approximate
each strip as a rectangle
with area a b
Note: This expression will be
non-negative even if the region
lies below the x-axis.
Lower curve

3. Area Between Curves
Suppose we want to know the area of a region that is bounded
above by one curve, y = f(x), and below by another, y = g(x):
Upper curve
We can approximate the
area of the region with the
Riemann sum a b
The limit of these sums as is
Lower curve

4. Definition: Area Between Curves
If f and g are continuous with throughout
[a, b], then the area between the curves y = f(x) and
y = g(x) from a to b is the integral of [f – g] from a to b,

5. Guided Practice Find the area of the region between and from to .
Now, use our new formula to
find the enclosed area:
First, graph the two curves
over the given interval:

6. Guided Practice Find the area of the region between and from to .
First, graph the two curves
over the given interval:

7. Guided Practice Find the area of the region enclosed by the parabola
and the line
The graph:
To find our limits of integration
(a and b), we need to solve
the system
Algebraically, or by calculator: a b

8. Guided Practice Find the area of the region enclosed by the parabola
and the line
Because the parabola lies above
the line, we have
The graph: a b
units
squared

9. Guided Practice Find the area of the region enclosed by the graphs of
and
The graph:
To find our limits of integration
(a and b), we need to solve
the system
Solve graphically: a b
Store the negative value as A
and the positive value as B.

10. Guided Practice Find the area of the region enclosed by the graphs of
and
Note: The trigonometric function
lies above the parabola…
The graph:
Let’s evaluate this one numerically…
Area: a b
units squared

11. Guided Practice
Find the area of the region R in the first quadrant that is bounded
above by and below by the x-axis and the line
The graph of R:
Area of region A: 2
(4,2) B 1 A 1 2 3 4

12. Guided Practice The graph of R: Area of region B: 2 (4,2) B 1 A 1 2 3

13. Guided Practice The graph of R: Area of R = Area of A + Area of B: 2
(4,2) B 1 A
Units
squared 1 2 3 4

14. Guided Practice
Find the area of the region enclosed by the given curves.
First, graph in [–2,12] by [0,3.5]
Intersection points:
Break into three subregions:

15. Guided Practice
Find the area of the region enclosed by the given curves.
First, graph in [–2,12] by [0,3.5]
Intersection points:
Break into three subregions: