Transcript Definite Integrals - West Virginia University

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 F0(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 s01 e2xdx.
 Solution: First, find an antiderivative of
e2x. This is F(x) = (1/2)e2x (why?). Now
compute that s01 e2xdx = F(1)-F(0) =
(1/2)e2 – (1/2)e0 = e2/2-1/2 = 3.1945…
