Riemann Sums & Definite Integrals

Finding Area with Riemann Sums Subintervals with equal width For convenience, the area of a partition is often divided into subintervals with equal width – in other words, the rectangles all have the same width. (see the diagram to the right and section 5-2)

Finding Area with Riemann Sums It is possible to divide a region into different sized rectangles based on an algorithm or rule (see graph above and example #1 on page 307)

Finding Area with Riemann Sums It is also possible to make rectangles of whatever width you want where the width and/or places where to take the height does not follow any particular pattern. Notice that the subintervals don’t seem to have a pattern. They don’t have to be any specific width or follow any particular pattern. Also notice that the height can be taken anywhere on each subinterval – not only at endpoints or midpoints!

What is a Riemann Sum Definition: Let f be defined on the closed interval [a,b], and let  be a partition [a,b] given by a = x0 < x1 < x2 < … < xn-1 < xn = b where xi is the width of the ith subinterval [xi-1, xi]. If c is any point in the ith subinterval, then the sum is called a Riemann sum of f for the partition .

New Notation for x b - a n When the partitions (boundaries that tell you where to find the area) are divided into subintervals with different widths, the width of the largest subinterval of a partition is the NORM of the partition and is denoted by |||| If every subinterval is of equal width, the partition is REGULAR and the norm is denoted by ||||= x = The number of subintervals in a partition approaches infinity as the norm of the partition approaches 0. In other words, |||| 0 implies that n  b - a n Is the converse of this statement true? Why or why not?

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.

Definite Integrals vs. Indefinite Integrals A definite integral is number. An indefinite integral is a family of functions. They may look a lot alike, however, definite integrals have limits of integration while the indefinite integrals have not limits of integration. Definite Integral Indefinite Integral

Theorem 5.4 Continuity Implies Integrability If a function f is continuous on the closed interval [a,b], then f is integrable on [a,b]. Is the converse of this statement true? Why or why not?

Evaluating a definite integral… To learn how to evaluate a definite integral as a limit, study Example #2 on p. 310.

Theorem 5.5 The Definite Integral as the Area of a Region If f is continuous and nonnegative on the closed interval [a,b], then the area of the region bounded by the graph of f , the x-axis, and the vertical lines x = a and x = b is given by Area =

Let’s try this out… Area = (base)(height) = (2)(4) = 8 un 2 Sketch the region Find the area indicated by the integral. Area = (base)(height) = (2)(4) = 8 un 2

Give this one a try… Area of a trapezoid = .5(width)(base1+ base2) Sketch the region Find the area indicated by the integral. base2 =5 Area of a trapezoid = .5(width)(base1+ base2) = (.5)(3)(2+5) = 10.5 un 2 base1=2 Width =3

Try this one… Area of a semicircle = .5( r2) = (.5)()(22) = 2  un 2 Sketch the region Find the area indicated by the integral. Area of a semicircle = .5( r2) = (.5)()(22) = 2  un 2

Properties of Definite Integrals If f is defined at x = a, then we define So, If f is integrable on [a,b], then we define So,

Additive Interval Property Theorem 5.6 Additive Interval Property If f is integrable on three closed intervals determined by a, b, and c, then

Properties of Definite Integrals Theorem 5.7 Properties of Definite Integrals If f and g are integrable on [a,b] and k is a constant, then the functions of kf and f  g are integrable on [a,b], and

HOMEWORK – yep…more practice Thursday, January 17 Read and take notes on section 5.3 and do p. 314 # 3, 6, 9, …, 45, 47, 49, 53, 65- 70 – Work on the AP Review Diagnostic Tests if you have time over the long weekend. Tuesday – January 22 – Get the Riemann sum program for your calculator and do p. 316 # 59-64, 71 and then READ and TAKE Notes on section 5-4 and maybe more to come!!