Definite Integrals Finding areas using the Fundamental Theorem of Calculus

The Definite Integral Definition: The definite integral of f(x) from x=a to x=b is Really, the definite integral computes the area under the curve by adding up the area of an ‘infinite’ number of rectangles

Riemann Sums Become The Definite Integral Increase the number of rectangle to get closer to the area under the curve

Computing Definite Integrals One approach is to compute a left or right Riemann sum for large numbers (~100) of rectangles. Computing Riemann Sums

A Better Way Of course, you wouldn’t want to do this a lot (unless you have to) The Fundamental Theorem of Calculus says that if you want to compute find a function F(x) so that F 0 (x) = f(x). Then

Using the FTC Any antiderivative F(x) will do so pick the one with C=0 Ex: Evaluate

Another Example Evaluate s 0 1 e 2x dx. Solution: First, find an antiderivative of e 2x. This is F(x) = (1/2)e 2x (why?). Now compute that s 0 1 e 2x dx = F(1)-F(0) = (1/2)e 2 – (1/2)e 0 = e 2 /2-1/2 = …