Calculus: Riemann Sums & Definite Integrals

1. Calculus:Riemann Sums & Definite Integrals Section 6.3

2. 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.

3. Finding Area with Riemann Sums • It is possible to divide a region into different sized rectangles based on an algorithm or rule

4. 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!

5. Riemann Sums Definition: Let f be defined on the closed interval [a,b], and let  be a partition [a,b] given bya = x0 < x1 < x2 < … < xn-1 < xn = bwhere 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 .

6. b - a n 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 |||| • 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 

7. 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.

8. 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

9. Theorem 6.4 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?

10. Theorem 6.5The 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 =

11. Let’s try this out… • Sketch the region • Find the area indicated by the integral. Area = (base)(height) = (2)(4) = 8 un2

12. 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

13. Sketch the region • Find the area indicated by the integral. Area of a semicircle = .5( r2) = (.5)()(22) = 2  un 2

14. 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,

15. Theorem 6.6 Additive Interval Property If f is integrable on three closed intervals determined by a, b, and c, then

16. Theorem6.7 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

17. Evaluating a definite integral… The Fundamental Theorem of Calculus If f is continuous at every point of , and if F is any antiderivative of f on , then