THE DEFINITE INTEGRAL

1
CHAPTER 4

• THE DEFINITE INTEGRAL

2
4.1 Introduction to Area

• Finding area of polygonal regions can be
  accomplished using area formulas for rectangles
  and triangles.
• Finding area bounded by a curve is more
  challenging.
• Consider that the area inside a circle is the
  same as the area of an inscribed n-gon where n is
  infinitely large.

3
Adding infinitely many terms together

• Summation notation simplifies representation
• Area under any curve can be found by summing
  infinitely many rectangles fitting under the
  curve.

4
4.2 The Definite Integral

• Riemann sum is the sum of the product of all
  function values at an arbitrary point in an
  interval times the length of the interval.
• Intervals may be of different lengths, the point
  of evaluation could be any point in the interval.
  
• To find an area, we must find the sum of
  infinitely many rectangles, each getting
  infinitely small.

5
Definition Definite Integral

• Let f be a function that is defined on the closed
  interval a,b.
• If exists, we say f
  is
• integrable on a,b. Moreover,
  called the definite integral
• (or Riemann integral) of f from a to be, is then
  given as that limit.

6
Area under a curve

• The definite integral from a to b of f(x) gives
  the signed area of the region trapped between the
  curve, f(x), and the x-axis on that interval.
• The lower limit of integration is a and the upper
  limit of integration is b.
• If f is bounded on a,b and continuous except at
  a finite number of points, then f is integrable
  on a,b. In particular, if f is continuous on
  the whole interval a,b, it is integrable on
  a,b.

7
Functions that are always integrable

• Polynomial functions
• Sin cosine functions
• Rational functions, provided that a,b contains
  no points where the denominator is 0.

8
4.3 First Fundamental Theorem

• Let f be continous on the closed interval a,b
  and let x be a (variable) point in (a,b). Then

9
What does this mean?

• The rate at which the area under the curve of
  function, f(t), is changing at a point is equal
  to the value of the function at that point.

10
4.4 The 2nd Fundamental Theorem of Calculus and
the Method of Substitution

• Let f be continuous (integrable) on a,b, and
  let F be any antiderivative of f on a,b. Then
  the definite integral is

11
Evaluate
12
Substitution Rule for Indefinite Integrals

• Let g be a differentiable function and suppose
  that F is an antiderivative of f. Then

13
What does this remind you of?

• It is the chain rule! (from differentiation)
• In this case, you have an integral with a
  function and its derivative both present in the
  integrand.
• This is often referred to as u-substitution
• Let ufunction and duthat functions derivative

14
Evaluate
15
Substitution Rule for Definite Integrals

• Let g have a continuous derivative on a,b, and
  let f be continuous on the range of g. Then
  where ug(x)

16
What does this mean?

• For a definite integral, when a substitution for
  u is made, the upper and lower limits of
  integration must change. They were stated in
  terms of x, they must be changed to be the
  corresponding values, in terms of u.
• When this change in the upper lower limits is
  made, there is no need to change the function
  back to be in terms of x. It is evaluated in
  terms of the upper lower limits in terms of u.

17
Evaluate
18
4.5 The Mean Value Theorem for Integrals and the
Use of Symmetry

• Average Value of a Function If f is integrable
  on the interval a,b, then the average value of
  f on a,b is

19
What does this mean?

• If you consider the definite integral from over
  a,b to be the area between the curve f(x) and
  the x-axis, f-average is the height of the
  rectangle that would be formed over that same
  interval containing precisely the same area.

20
Mean Value Theorem for Integrals

• If f is continuous on a,b, then there is a
  number c between a and b such that

21
Symmetry Theorem

• If f is an even function then
• If f is an odd function, then

22
4.6 Numerical Integration

• If f is continuous on a closed interval a,b,
  then the definite integral must exist. However,
  it is not always easy or possible to find the
  definite integral.
• In these cases, we use other methods to closely
  approximate the definite integral.

23
Methods for approximating a definite integral

• Left (or right or midpoint) Riemann sums
  (estimate the area with rectangles)
• Trapezoidal Rule (estimate with several
  trapezoids)
• Simpsons Rule (estimate the area with the region
  contained under several parabolas)

24
Summary of numerical techniques

• Approximating the definite integral of f(x) over
  the interval from a to b.

25
(No Transcript)