AP CALCULUS ABChapter 7:

Applications of Definite IntegralsSection 7.2:

Areas in the Plane

What you’ll learn about Area Between Curves Area Enclosed by Intersecting Curves Boundaries with Changing Functions Integrating with Respect to y Saving Time with Geometric Formulas

…and whyThe techniques of this section allow us to

compute areas of complex regions of the plane.

Area Between Curves

Partition the region into vertical strips of equal width .

Each rectangle has area for some in its

respective subinterval. Approximate the area of each region

with the Riemann sum

k k k

k

x

f c g c x c

f c g c

.k

x

The limit of these sums as 0 is .b

ax f x g x dx

Section 7.2 – Areas in the Plane 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,

f x g x

.b

aA f x g x dx

Example Applying the Definition

Find the area of the region between cos and sin

from 0 to .3

y x y x

x x

30

3

0sin cos

Note that cos is above sin in the given interval.

cos - sin

3 1 units squared.

2

x x

y x y x

A x x dx

Section 7.2 – Areas in the Plane Example:

2 23

3

1sec 4sin

2

4.189

A t t dt

Section 7.2 – Areas in the Plane Area Enclosed by Intersecting Curves:

When a region is enclosed by intersecting curves, the intersection points give the limits of integration.

Example Area of an Enclosed Region

2Find the area of the region enclosed by the parabola 1

and 1.

y x

y x

2

Graph the curves to view the region.

The limits of integration are found by solving the equation

1 1. The solutions are -1 and 2.

Since the line lies above the parabola on -1,2 , the area

integra

x x x x

2

2 2

1

2

-1

3 2

23 2

nd is 1 1 .

2

9 units squared.

2

x xx

x x

A x x dx

Section 7.2 – Areas in the Plane Example:

2 2

1

2 2

1

2 2

1

23 2

1

3 23 2

4 2

4 2

2

23 2

1 12 22 2 2 1

3 2 3 2

8 1 14 2 2

3 3 2

8 1 14 2 2

3 3 21

8 32

5.5

A x x dx

x x dx

x x dx

x xx

Section 7.2 – Areas in the Plane Boundaries with Changing Functions:

If a boundary of a region is defined by more than one function, we can partition the region into sub-regions that correspond to the function changes and proceed as usual.

Section 7.2 – Areas in the Plane Example:

1 22

0 1

1 23 2

0 1

2 2

2

23 2

1 0 2 12 2 2 1

3 3 2 2

1 14 2 2

3 22 3 5

6 6 6

A x dx x dx

x xx

Section 7.2 – Areas in the Plane Integrating with Respect to y:

Sometimes the boundaries of a region are more easily described by functions of y than by functions of x. We can use approximating rectangles that are horizontal rather than vertical and the resulting basic formula has y in place of x.

Integrating with Respect to y

If the boundaries of a region are more easily described by

functions of , use horizontal approximating rectangles.y

Example Integrating with Respect to y 2Find the area of the region bounded by the curves 1

and 1.

x y

y x

2

1 2

0

1 2

0

1

0

2 3

2 3

Solve for in terms of : 1 and 1. Find the points

of intersection: -1,0 and 0,1 .

1 1

1

6

y y

x y x y x y

A y y dy

y y dy

Section 7.2 – Areas in the Plane Example

1 2 3 2

0

1 2 3

0

13 4 2

0

12 12 2 2

10 12 2

10 12 2

3 4 2

103 1

3

10 6 4

3 3 3

A y y y y dy

y y y dy

y y y

Section 7.2 – Areas in the Plane Saving Time with Geometry Formulas:

Sometimes you can save time by realizing when you have common geometric figures for all or part of your area, and you can use your area formulas for these figures.

Example Using Geometry Find the area of the region enclosed by the graphs of 1,

1 and the -axis.

y x

y x x

3

-1

3

1

3

22

13

Find the area under the curve 1 over the interval -1,3

and subtract the area of the triangle:

1 1 2 2

2

2

10 units squared

3

x

y x

A x dx

Section 7.2 – Areas in the Plane Example:

1 2

0

13

0

11 1

2

1

3 2

1 0 1

3 3 2

2 3 5

6 6 6

A x dx

x

FRQ Problem 2007B1