7.2 Areas in the Plane.

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Presentation transcript:

7.2 Areas in the Plane

Quick Review

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? http://www.math.psu.edu/dlittle/java/calculus/areabetweencurves.html

Area Between Curves

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

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

Integrating with Respect to y

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

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

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