1. Chapter 5: The Definite Integral Section 5.2: Definite Integrals
AP CALCULUS AB
Chapter 5:
The Definite Integral
Section 5.2:
Definite Integrals

2. What you’ll learn about
Riemann Sums
The Definite Integral
Computing Definite Integrals on a Calculator
Integrability
… and why
The definite integral is the basis of integral calculus, just as the derivative is the basis of differential calculus.

3. Sigma Notation

4. Section5.2 – Definite Integrals
Definition of a Riemann Sum
f is defined on the closed interval [a, b], and is a partition of [a, b] given by
where is the length of the ith subinterval.
If ci is any point in the ith subinterval, then the sum
is called a Riemann Sum of f for the partition .
a b
Partitions do not have to
be of equal width
If the are of equal width,
then the partition is
regular and

5. The Definite Integral as a Limit of Riemann Sums

6. The Existence of Definite Integrals

7. The Definite Integral of a Continuous Function on [a,b]

8. The Definite Integral

9. Section 5.2 – Definite Integrals
If f is defined on the closed interval [a, b] and the limit
exists, then f is integrable on [a, b] and the limit is denoted by
The limit is called the definite integral of f from a to b. The number a is the lower limit of integration, and the number b is the upper limit of integration.
The function is the integrand
x is the variable of
integration

10. Example Using the Notation

11. Section 5.2 – Definite Integrals
Theorem: If y=f(x) is
nonnegative and integrable
over a closed interval [a, b],
then the area under the curve
y=f(x) from a to b is the
integral of f from a to b,
If f(x)< 0, from a to b (curve is under the x-axis),
then
a b

12. Area Under a Curve (as a Definite Integral)

13. Area

14. The Integral of a Constant

15. Section 5.2 – Definite Integrals
To find Total Area Numerically (on the calculator)
To find the area between the graph of y=f(x) and the x-axis over the interval
[a, b] numerically, evaluate:
On the TI-89:
nInt (|f(x)|, x, a, b)
On the TI-83 or 84:
fnInt (|f(x)|, x, a, b) Note: use abs under Math|Num for absolute value

16. Example Using NINT (FnInt)

17. Example Using NINT (FnInt)

18. Discontinuous Functions
The Reimann Sum process guarantees that all functions that are continuous are integrable. However, discontinuous functions may or may not be integrable.
Bounded Functions: These are functions with a top and bottom, and a finite number of discontinuities on an interval [a,b]. In essence, a RAM is possible, so the integral exists, even if it must be calculated in pieces. A good example from the Finney book is f(x) = |x|/x.

19. Discontinuous Functions
An example of a discontinuous function (badly discontinuous), which is also known as a non-compact function, is given also:
This function is 1 when x is rational, zero when x is irrational. On any interval, there are an infinite number of rational and irrational values.