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 Simplified Reminder Version Definition of a Definite Integral n b lim f xk x f x dx n k 1 where a b a x n xk a k x 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 n n Ak fx k x kf g x kgx k x kg x k x x f lim n k 1 k 1 Ak f xk g xk xk x b f x g x dx a Example 1 Find the area bounded by f x 1 ex 1 and g x 3cos x over 0 2 2 A 3 cos x 1 e x 1 dx 0 3 sin x x e 3 e 2 2 831 x 1 2 0 2 1 1 e No interval given Example 2 Find the area bounded by f x x2 and g x x3 Find the points of intersection 1 A x x dx 2 3 0 1 x x 3 4 0 3 4 1 1 1 3 4 12 Functions that cross each other Example 3 Find the area bounded by f x 12 3x and g x x3 4x2 4x 16 Find the points of intersection 1 A x 3 4 x 2 4 x 16 12 3 x dx 1 4 12 3x x3 4 x 2 4 x 16 dx 1 15 1 1 9 1 4 0 x3 4 x 2 x 4 dx 1 4 3 2 x 4 x x 4 dx 1 253 21 08333 12 Example 4 Regions bounded by more than two functions Find the area bounded by f x 3x 6 g x 2 x and h x x 2 2 A 12 x 2 2 x dx 0 h x 3 12 x 2 3 x 6 dx 2 f x g x x dx 8 2 3 3 0 2 x 3 4 5 2 2 8x 2 2 0 3 0 24 5 4 x 45 4 2 x dx 3 2 19 16 5 4 Example 5 Regions bounded relations i e not functions Find the area inside the circle by x2 y2 9 and above the line 3x 2y 6 90 0 36 132 2 A 2 9 9 x x 9dx x 29 x 322 xdx 3 dx 23 3 0 36 13 9 x 2 32 x 3 dx 0 9 9 1 12 54 sin 23 583 2 2 13 13 Example 6 Regions bounded functions of y Find the area bounded by the parabolas x y2 9 and x y2 0 92 A 2 0 y x y 9 x 9 x dx 2 9 x dx 92 Example 6 Regions bounded functions of y Find the area bounded by the parabolas x y2 9 and x y2 0 A 2 2 9 y 3 2 2 9 2 y dy 9 y 2 3 3 0 x y2 A 2 0 y 2 dy x 9 y 2 0 92 2 9 x dx 2 9 x dx 92 2 y 3 3 0 2 12 7279