AP CALCULUS AB Chapter 7: Applications of Definite Integrals Section 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 why The techniques of this section allow us to compute areas of complex regions of the plane.

Area Between Curves

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,

Example Applying the Definition

Section 7.2 – Areas in the Plane  Example:

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

Section 7.2 – Areas in the Plane  Example:

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:

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

Example Integrating with Respect to y

Section 7.2 – Areas in the Plane  Example

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

Section 7.2 – Areas in the Plane  Example:

FRQ Problem  2007B1