AP CALCULUS AB

1
AP CALCULUS AB

• Chapter 7
• Applications of Definite Integrals
• Section 7.2
• Areas in the Plane

2
What youll 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 why
• The techniques of this section allow us to
  compute areas of complex regions of the plane.

3
Area Between Curves

4
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 yf(x) and yg(x) from a to b is the
  integral of f g from a to b,

5
Example Applying the Definition

6
Section 7.2 Areas in the Plane

• Example

7
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.

8
Example Area of an Enclosed Region

9
Section 7.2 Areas in the Plane

• Example

10
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.

11
Section 7.2 Areas in the Plane

• Example

12
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.

13
Integrating with Respect to y

14
Example Integrating with Respect to y

15
Section 7.2 Areas in the Plane

• Example

16
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.

17
Example Using Geometry

18
Section 7.2 Areas in the Plane

• Example

19
FRQ Problem

• 2007B1