1. Chapter 6 The Integral Sections 6.1, 6.2, and 6.3

2. The Integral The Indefinite Integral Substitution
The Definite Integral As a Sum
The Definite Integral As Area

3. Introduction
A physicist who knows the velocity of a particle might wish to know its position at a given time.
A biologist who knows the rate at which a bacteria population is increasing might want to deduce what the size of the population will be at some future time.

4. Antiderivatives Definition
In each case, the problem is to find a function F whose derivative is a known function f.
If such a function F exists, it is called an antiderivative of f.
Definition
A function F is called an antiderivative of f on
an interval I if F’(x) = f (x) for all x in I.

5. Antiderivatives For instance, let f (x) = x2.
It is not difficult to discover an antiderivative of f if we keep the Power Rule in mind.
In fact, if F(x) = ⅓ x3, then F’(x) = x2 = f (x).

6. Antiderivatives
However, the function G(x) = ⅓ x also satisfies G’(x) = x2.
Therefore, both F and G are antiderivatives of f.
Indeed, any function of the form H(x)=⅓ x3 + C, where C is a constant, is an antiderivative of f.
The question arises: Are there any others?

7. Antiderivatives
Theorem
If F is an antiderivative of f on an interval I, then the most general antiderivative of f on I is F(x) + C
where C is an arbitrary constant.

8. Antiderivatives
Going back to the function f (x) = x2, we see that the general antiderivative of f is ⅓ x3 + C.

9. Family of Functions
By assigning specific values to C, we obtain a family of functions.
Their graphs are vertical
translates of one another.
This makes sense, as each
curve must have the same
slope at any given value
of x.

10. Notation for Antiderivatives
The symbol is traditionally used to represent the most general an antiderivative of f on an open interval and is called the indefinite integral of f .
Thus, means F’(x) = f (x)

11. Indefinite Integral The expression:
read “the indefinite integral of f with respect to x,” means to find the set of all antiderivatives of f.
x is called the variable of integration
Integrand
Integral sign

12. Indefinite Integral For example, we can write
Thus, we can regard an indefinite integral as representing an entire family of functions (one antiderivative for each value of the constant C).

13. Constant of Integration
Every antiderivative F of f must be of the form F(x) = G(x) + C, where C is a constant.
Example:
Represents every possible antiderivative of 6x.

14. Power Rule for the Indefinite Integral
Example:

15. Power Rule for the Indefinite Integral
Indefinite Integral of ex and bx

16. Sum and Difference Rules
Example:

17. Constant Multiple Rule
Example:

18. Example - Different Variable
Find the indefinite integral:

19. Position, Velocity, and Acceleration Derivative Form
If s = s(t) is the position function of an object at time t, then
Velocity = v =
Acceleration = a =
Integral Form

20. Integration by Substitution
Method of integration related to chain rule. If u is a function of x, then we can use the formula

21. Integration by Substitution
Example: Consider the integral:
Sub to get
Integrate
Back Substitute

22. Example: Evaluate
Pick u, compute du
Sub in
Integrate
Sub in

23. Example: Evaluate

24. Example: Evaluate

25. The Definite Integral
Let f be a continuous function on [a, b]. If F is any antiderivative of f defined on [a, b], then the definite integral of f from a to b is defined by
is read “the integral, from a to b of f (x) dx.”

26. Notation In the notation , f (x) is called the integrand.
a and b are called the limits of integration; a is the lower limit and b is the upper limit.
For now, the symbol dx has no meaning by itself; is all one symbol. The dx simply indicates that the independent variable is x.

27. The Definite Integral
The procedure of calculating an integral is called integration. The definite integral is a number. It does not depend on x.
Also note that the variable x is a “dummy variable.”

28. Geometric Interpretation of the Definite Integral
The Definite Integral As Area
The Definite Integral As Net Change of Area

29. Definite Integral As Area
If f is a positive function defined for a ≤ x ≤ b, then the definite integral represents the area under the curve y = f (x) from a to b

30. Definite Integral As Area
If f is a negative function for a ≤ x ≤ b, then the area between the curve y = f (x) and the x-axis from a to b, is the negative of

31. Definite Integral As Area
Consider y = f (x) = 0.5x + 6 on the interval [2,6] whose graph is given below,

32. Definite Integral As Area
Consider y = f (x) = 0.5x + 6 on the interval [2,6] whose graph is given below,

33. Definite Integral as Net Area
If f changes sign on the interval a ≤ x ≤ b, then definite integral represents the net area, that is, a difference of areas as indicated below:
a b R1 R2 R3
Area of R1 – Area of R2 + Area of R3

34. Total Area
If f changes sign on the interval a ≤ x ≤ b, and we need to find the total area between the graph and the x-axis from a to b, then
Total Area  Area of R1 + Area of R2 + Area of R3
Area of R1
a b R1 R2 R3
Area of R2 c d
Area of R3

35. Area Using Geometry Example: Use geometry to compute the integral–1 5
Area = 8
Area = 2

36. Area Using Antiderivatives
Example: Use an antiderivative to compute the integral
First, we need an antiderivative of

37. Area Using Antiderivatives
Example: Now find the total area bounded by the curve and the x-axis from x  –1 to x  5. R2 –1 1 5 R1
Total Area  Area of R1 + Area of R2

38. R2 –1 1 5 R1
Total Area   10
Area of R1
Area of R2

39. Evaluating the Definite Integral
Example: Calculate

40. Substitution for Definite Integrals
Example: Calculate
Notice limits change

41. Computing Area
Example: Find the area enclosed by the x-axis, the vertical lines x = 0, x = 2 and the graph of
Gives the area since 2x3 is nonnegative on [0, 2].
Antiderivative

42. The Definite Integral As a Total
If r (x) is the rate of change of a quantity Q (in units of Q per unit of x), then the total or accumulated change of the quantity as x changes from a to b is given by

43. The Definite Integral As a Total
Example: If at time t minutes you are traveling at a rate of v(t) feet per minute, then the total distance traveled in feet from minute 2 to minute 10 is given by