1. Applications of Definite Integrals
AP CALCULUS AB
Chapter 7:
Applications of Definite Integrals
Section 7.2:
Areas in the Plane

2. 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 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 y=f(x) and y=g(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