Section 5.2: Definite Integrals Objectives: Define a Riemann Sum Connect Riemann Sum and Definite Integral Relate the Definite Integral and Area under the curve

Sigma Notation k tells us where to begin, n tells us where to end If n is ∞, terms go on forever, and ever, and ever, and ever……

Reimann Sum We can use sigma notation to approximate the area under a curve We will add up all the areas of the tiny, little rectangles. We call this a Reimann Sum Rectangles can lie either above or below the x-axis

The Definite Integral as a Limit of Riemann Sums f(x) is on a closed interval [a,b] f is integrable on [a,b] and is the definite integral of f over [a,b] NOTES: is called the partition, and is the longest subinterval length (also may see written as ) is the height of the rectangle (it is the value of the function at some value c in the kth subinterval is the width of the rectangle.

The Definite Integral of a Continuous Function of [a, b] Let f be continuous [a, b] be partitioned into n subintervals of equal length Δx = (b – a)/n. Then the definite integral of f over [a, b] is given by where each ck is chosen arbitrarily in the kth subinterval. (the more subintervals you have, the more accurate the area)

The Existence of Definite Integrals 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.

Definite Integral notation When you find the value of the integral, you have evaluated the integral. The definite integral is a number!!

Let’s break it down….. What does all this mean???? Upper limit of Integration integrand x is the variable of integration Integral Sign Lower limit of integration Read as “The integral from a to b of f of x dx”

Express the limit as an integral. on [0,4]

Definite Integral and Area Area Under a Curve (as a Definite Integral) 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 be is the integral of f from a to b.

Non-positive Integrable Functions

Any Integrable Function = (area above the x-axis) – (area below the x-axis)

Using Geometric Formulas to evaluate the integral

The Integral of a Constant If f(x) = c, where c is a constant, on the interval [a, b], then

Examples: Using Geometric Formulas

If you were driving at a constant speed of 65 mph from 8am to 11 am, how far did you travel? Write a definite integral, and evaluate.

Discontinuous Integrable Functions: Definition implies continuity, but there are some discontinuous integrable functions.