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Section 5.2: Definite Integrals

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1 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

2 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……

3 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

4 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.

5 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)

6 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.

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

8 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”

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

10 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.

11 Non-positive Integrable Functions

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

13 Using Geometric Formulas to evaluate the integral

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

15 Examples: Using Geometric Formulas




19 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.

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

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