Riemann Sums & Definite Integrals Section 5.3

Finding Area with Riemann Sums 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)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) Subintervals with equal width

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)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.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 = x 0 < x 1 < x 2 < … < x n-1 < x n = b where  x i is the width of the ith subinterval [x i-1, x i ]. 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 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 ||  ||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 =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 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 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 definite integrals have limits of integration while the indefinite integrals have not limits of integration. 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… 1.Sketch the region 2.Find the area indicated by the integral. Area = (base)(height) = (2)(4) = 8 un 2

Give this one a try… 1.Sketch the region 2.Find the area indicated by the integral. Area of a trapezoid =.5(width)(base1+ base2) = (.5)(3)(2+5) = 10.5 un 2 Width =3 base1=2 base2 =5

Try this one… 1.Sketch the region 2.Find the area indicated by the integral. Area of a semicircle =.5(  r 2 ) = (.5)(  )(2 2 ) = 2  un 2

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

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

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 Theorem 5.7

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, – 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!!