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

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.

Sigma Notation

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

The Definite Integral as a Limit of Riemann Sums

The Existence of Definite Integrals

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

The Definite Integral

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

Example Using the Notation

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

Area Under a Curve (as a Definite Integral)

Area

The Integral of a Constant

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

Example Using NINT (FnInt)

Example Using NINT (FnInt)

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.

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.