The Definite Integral

1
Lesson 5-2

• The Definite Integral

2
Ice Breaker

• Find area between x-axis and y 2x on 0,3
  using 3 rectangles and right-end points

Area of Rectangle h w
xi a i?x RH w ?x (b-a)/n
(3-0) / 3 1
h f(xi) 2(xi)
R1 (2(x1)) (?x) 2(1) 1 2
R2 (2(x2)) (?x) 2(2) 1 4
R3 (2(x3)) (?x) 2(3) 1 6
Area of Triangle ½ (b)(h)
½ (3)(6) 9
RH 1 (246) 14
3
Riemann Sums

• Let f be a function that is defined on the closed
  interval a,b. If ? is a partition of a,b and
  ?xi is the width of the ith interval, ci, is any
  point in the subinterval, then the sum
• f(ci)?xi is called a Riemann Sum of f.
  Furthermore,
• if exists, lim f(ci)?xi we say f is
  integrable on a,b.
• The definite integral, f(x)dx , is the area
  under the curve

n?8
4
Definite Integral vs Riemann Sum
3i
3 Area Lim ? 3(2 -----) 8 (---)
n n
5
Summary Homework

• Summary
• Riemann Sums are Limits of Infinite sums
• Riemann Sums give exact areas under the curve
• Riemann Sums are the definite integral
• Homework
• pg 390 - 393 3, 5, 9, 17, 20, 33, 38