The Definite Integral

1
The Definite Integral
2
Riemann Sums

• Sigma notation enables us to express a large sum
  in compact form

3
Riemann Sums

• LRAM, MRAM,and RRAM are examples of Riemann sums
• Sn
• This sum, which depends on the partition P and
  the choice of the numbers ck,is a Riemann sum for
  f on the interval a,b

4
Definite Integral as a Limit of Riemann Sums

• Let f be a function defined on a closed interval
  a,b. For any partition P of a,b, let the
  numbers ck be chosen arbitrarily in the
  subintervals xk-1,xk.
• If there exists a number I such that
• no matter how P and the cks are chosen, then f
  is integrable on a,b and I is the definite
  integral of f over a,b.

5
Definite Integral of a continuous function on
a,b

• Let f be continuous on a,b, and let a,b be
  partitioned into n subintervals of equal length
  ?x (b-a)/n. Then the definite integral of f
  over a,b is given by
• where each ck is chosen arbitrarily in the kth
  subinterval.

6
Definite integral

• This is read as the integral from a to b of f of
  x dee x or sometimes as the integral from a to
  b of f of x with respect to x.

7
Using Definite integral notation
The function being integrated is f(x) 3x2 2x
5 over the interval -1,3
8
Definition Area under a curve

• If y f(x) is nonnegative and integrable over a
  closed interval a,b, then the area under the
  curve of y f(x) from a to b is the integral of
  f from a to b,

9
Nonpositive regions

• If the graph is nonpositive from a to b then

10
Area of any integrable function

• (area above the x-axis)
• (area below x-axis)

11
Integral of a Constant

• If f(x) c, where c is a constant, on the
  interval a,b, then

12
Evaluating Integrals using areas

• We can use integrals to calculate areas and we
  can use areas to calculate integrals.
• Using areas, evaluate the integrals
• 1)
• 2)

13
Evaluating Integrals using areas

• Evaluate using areas
• 3)
• 4) (altb)

14
Evaluating integrals using areas

• Evaluate the discontinuous function
• Since the function is discontinuous at x 0, we
  must divide the areas into two pieces and find
  the sum of the areas
• -1 2 1

15
Integrals on a Calculator

• You can evaluate integrals numerically using the
  calculator. The book denotes this by using NINT.
  The calculator function fnInt is what you will
  use.
• fnInt(xsinx,x,-1,2) is
  approx. 2.04

16
Evaluate Integrals on calculator

• Evaluate the following integrals numerically
• approx. 3.14
• approx. .89

17
Rules for Definite Integrals

• Order of Integration

18
Rules for Definite Integrals

• Zero

19
Rules for Definite Integrals

• 3) Constant Multiple

20
Rules for Definite Integrals

• 4) Sum and Difference

21
Rules for Definite Integrals

• 5) Additivity

22
Rules for Definite Integrals

• Max-Min Inequality If max f and min f are the
  maximum and minimum values of f on a,b then
• min f (b a) max f
  (b a)

23
Rules for Definite Integrals

• Domination f(x) g(x) on a,b
• f(x) 0 on a,b 0

24
Using the rules for integration

• Suppose
• Find each of the following integrals, if
  possible
• b) c)
• d) e) f)

25
Using the rules for definite integrals

• Show that the value of
• is less than 3/2
• The Max-Min Inequality rule says the
• max f . (b a) is an upper bound.
• The maximum value of v(1cosx) on 0,1 is v2 so
  the upper bound is
• v2(1 0) v2 , which is less than
  3/2

26
Average (Mean) Value

• If f is integrable on a,b, its average (mean)
  value on a,b is
• av(f)
• Find the average value of f(x) 4 x2 on 0,3
  . Does f actually take on this value at some
  point in the given interval?

27
Applying the Mean Value

• Av(f)
• 1/3(3) 1
• 4 x2 1 when x v3 but only v3 falls in the
  interval from 0,3, so x v3 is the place where
  the function assumes the average.

28
Mean Value Theorem for Definite Integrals

• If f is continuous on a,b, then at some point c
  in a,b,

29
The Fundamental Theorem of Calculus, Part I

• If f is continuous on a,b, then the function
• F(x)
• has a derivative at every point x in a,b, and

30
Applications of The Fundamental Theorem of
Calculus, Part I

• 1.
• 2.
• 3.

31
Applications of The Fundamental Theorem of
Calculus, Part I

• Find dy/dx.
• y
• Since this has an x on both ends of the integral,
  it must be separated.

32
Applications of The Fundamental Theorem of
Calculus, Part I

33
Applications of The Fundamental Theorem of
Calculus, Part I

34
The Fundamental Theorem of Calculus, Part 2

• If f is continuous at every point of a,b, and
  if F is any antiderivative of f on a,b, then
• This part of the Fundamental Theorem is also
  called the Integral Evaluation Theorem.

35
Trapezoidal Rule

• To approximate , use
• T (y0 2y1 2y2 . 2yn-1 yn)
• where a,b is partitioned into n
  subintervals of equal length h (b-a)/n.

36
Using the trapezoidal rule

• Use the trapezoidal rule with n 4 to estimate
• h (2-1)/4 or ¼, so
• T 1/8( 12(25/16)2(36/16)2(49/16)4)
• 75/32 or about 2.344

37
Simpson Rule

• To approximate , use
• S (y0 4y1 2y2 4y3. 2yn-2 4yn-1
  yn)
• where a,b is partitioned into an even number
  n subintervals of equal length h (b a)/n.

38
Using Simpsons Rule

• Use Simpsons rule with n 4 to estimate
• h (2 1)/4 ¼, so
• S 1/12 (1 4(25/16) 2(36/16) 4(49/16) 4)
• 7/3