Definite Integrals

1
Definite Integrals

• Finding areas using the Fundamental Theorem of
  Calculus

2
The Definite Integral

• Definition The definite integral of f(x) from
  xa to xb is
• Really, the definite integral computes the area
  under the curve by adding up the area of an
  infinite number of rectangles

3
Riemann Sums Become The Definite Integral

• Increase the number of rectangle to get closer to
  the area under the curve

4
Computing Definite Integrals

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

5
A Better Way

• Of course, you wouldnt 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

6
Using the FTC

• Any antiderivative F(x) will do so pick the one
  with C0
• Ex Evaluate

7
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