On-line Math 21

5.2  The Fundamental Theorem of Calculus

Example 5

ó õ 5 2  f(x)dx.

where

f(x) = ì í î 3x-4, if x ³ 4 x+2, if x < 4

Solution

Split the integral into regions where we can unambiguously describe the function:

ó õ 5 2  f(x)dx = ó õ 4 2  f(x)dx+ ó õ 5 4  f(x)dx = ó õ 4 2  x+2dx+ ó õ 5 4  3x-4dx = x2/2+2x|24+3x2/2-4x|45 = (42/2+2(4))-(22/2+2(2))+(3(52)/2-4(5))-(3(42)/2-4(4)) = 39/2.

Verify for yourself that the antidifferentiation is correct..

Copyright (c) 2000 by David L. Johnson.