MTH 252 Integral Calculus Chapter 7 – Applications of the Definite Integral Section 7.1 – Area Between Two Curves Copyright © 2006 by Ron Wallace, all rights reserved.

Reminder: Definition of a Definite Integral Simplified Version where …

Area Between Curves Given that f(x) and g(x) are continuous over [ a,b ] and f(x)  g(x) when x  [ a,b ], find the area bounded by f(x), g(x), x = a, and x = b.

Area Between Curves Given that f(x) and g(x) are continuous over [ a,b ] and f(x)  g(x) when x  [ a,b ], find the area bounded by f(x), g(x), x = a, and x = b.

Example 1 Find the area bounded by f(x) = 1 - e x-1 and g(x) = 3cos x over [ 0,  /2 ].

Example 2 No interval given. Find the area bounded by f(x) = x 2 and g(x) = x 3. Find the points of intersection!

Example 3 Functions that cross each other. Find the area bounded by f(x) = x and g(x) = x 3 – 4x 2 – 4x Find the points of intersection!

Example 4 Find the area bounded by f(x) = 3x - 6, g(x) = 2 - x, and h(x) = ½ x + 2. Regions bounded by more than two functions.

Example 5 Regions bounded relations (i.e. not functions). Find the area inside the circle by x 2 + y 2 = 9, and above the line 3x – 2y = 6.

Example 6 Regions bounded functions of y. Find the area bounded by the parabolas x + y 2 = 9, and x – y 2 = 0

Example 6 Regions bounded functions of y. Find the area bounded by the parabolas x + y 2 = 9, and x – y 2 = 0