Riemann sums, the definite integral, integral as area Section 5.2a

First, we need a reminder of sigma notation: How do we evaluate: …and what happens if an “infinity” symbol appears above the sigma???  The terms go on indefinitely!!! LRAM, MRAM, and RRAM are all examples of Riemann sums, because of how they were constructed. In this section, we start with a more general account of these sums………………….observe…………….

We start with an arbitrary function f(x), defined on a closed interval [a, b]. Partition the interval [a, b] into n subintervals by choosing n – 1 points between a and b, subject only to a b Letting a = x and b = x , we have a partition of [a, b]: n

We start with an arbitrary function f(x), defined on a closed interval [a, b]. The partition P determines n closed subintervals. The subinterval is , which has length x In each subinterval we choose some number, denoting the number chosen from the subinterval by . On each subinterval, we create a rectangle that reaches from the x-axis to touch the curve at .

On each subinterval, we create a rectangle that reaches from the x-axis to touch the curve at . a b On each subinterval, we form the product (which can be positive, negative, or zero…) Area of each rectangle!!!

Riemann sum for f on the interval [a, b] Finally, take the sum of these products: This sum is called the Riemann sum for f on the interval [a, b]

Riemann Sums As with LRAM, MRAM, and RRAM, all Riemann sums for a given interval [a, b] will converge to common value, as long as the subinterval lengths all tend to zero. To ensure this last condition, we require that the longest subinterval (called the norm of the partition, denoted ||P||) tends to zero…

Definition: The 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 numbers be chosen arbitrarily in the subintervals . If there exists a number I such that no matter how P and the ‘s are chosen, then f is integrable on [a, b] and I is the definite integral of f over [a, b].

Theorem: The Existence of Definite Integrals In particular, if f is continuous, then choices about partitions and ‘s don’t matter, as long as the longest subinterval tends to zero: All continuous functions are integrable. That is, if a function f is continuous on an interval [a, b], then its definite integral over [a, b] exists. This theorem allows for a simpler definition of the definite integral for continuous functions. We need only consider the limit of regular partitions (in which all subintervals have the same length)…

The 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 . Then the definite integral of f over [a, b] is given by where each is chosen arbitrarily in the subinterval.

Integral Notation The Greek “S” is changed to an elongated Roman “S,” so that the integral retains its identity as a “sum.” This is read as “the integral from a to b of f of x dee x” or “the integral from a to b of f of x with respect to x”

Integral Notation The function is Upper limit the integrand of integration x is the variable of integration (also called a dummy variable) Integral Sign Lower limit of integration When you find the value of the integral, you have evaluated the integral Integral of f from a to b

A Quick Practice Problem The interval [–1, 3] is partitioned into n subintervals of equal length . Let denote the midpoint of the subinterval. Express the given limit as an integral. The function being integrated is over the interval [–1, 3]...

Definition: Area Under a Curve (as a Definite Integral If is nonnegative and integrable over a closed interval [a, b], then the area under the curve of from a to b is the integral of from a to b,

Practice Problem Evaluate the integral What is the graph of the integrand??? From Geometry-Land: (0, 2) Area = (–2, 0) (2, 0)

What happens when the curve is below the x-axis?  The area is negative!!! Area = when

For any integrable function, If an integrable function y = f (x) has both positive and negative values on the interval [a, b], add the areas of the rectangles above the x-axis, and subtract those below the x-axis: For any integrable function, = (area above x-axis) – (area below x-axis)

What happens with constant functions? If f (x) = c, where c is a constant, on the interval [a, b], then Does this make sense graphically??? Quick Example:

Practice Problems Use the graph of the integrand and areas to evaluate the given integral.

Practice Problems Use the graph of the integrand and areas to evaluate the given integral.

Practice Problems Use the graph of the integrand and areas to evaluate the given integral. 3b 3a a b