1. 7.2
Areas in the Plane

6. Quick Review

7. 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
Essential Question
How do we use integrals to compute areas
of complex regions of the plane?

8. Area Between Curves

9. Area Between Curves Example Applying the Definition units squared
If f and g are continuous with f (x) > g(x) 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
Find the area of the region between y = cos x and y = sin x from x = 0 to x = p/4.
units squared

10. Example Area of an Enclosed Region
Find the area of the region enclosed by the parabola y = x2 – 1 and y = x + 1.
Graph the curve to view the region.
Solve the equation to find the interval.
units squared

11. Integrating with Respect to y

12. Example Integrating with Respect to y
Find the area of the region bounded by the curves x = y2 – 1 and y = x + 1.
Graph the curve to view the region.
Solve for x in terms of y.
Points of intersection:
units squared

13. Example Using Geometry
Find the area of the region enclosed by the graphs and y = x – 1 and the x-axis.
Graph the curve to view the region.
Find the area under the curve over the interval [-1, 3].
Then subtract the triangle.
units2

14. Pg. 395, 7.2 #1-10 all
Pg. 396, 7.2 #11-41 odd