The document introduces Riemann sums and the definite integral. It states that taking the limit of Riemann sums as the number of rectangles approaches infinity converges to the definite integral, representing the exact area under a curve. It provides examples of calculating integrals using a graphing calculator and comparing to Riemann sums. Common integrals like a constant or linear function relate to geometric shapes like rectangles or triangles.

1. Introduction to the Definite Integral Unit 6 Lesson 2 E. Alexander Burt Potomac School

2. Riemann Sums We can approximate the area under a curve with rectangles. Each rectangle has a base of D x and a height of f(x)

3. The area under the curve is approximately the sum of the area of those rectangles

4. The more rectangles we sum (and the smaller D x gets) the closer our approximation will get to the actual area.

5. Riemann Sums A larger number of sub-intervals results in a closer approximation. Note that all the approximations approach a specific value. http://commons.wikimedia.org/wiki/File:Riemann_sum_convergence.png

6. The Definite Integral If we had an infinite number of rectangles of width dx, the Riemann sum would “converge” to a specific value – the “definite integral” of f(x) for the interval a to b

7. Calculus Lawyer Talk: if the function is continuous from a to b, the definite integral over the interval a to b exists.

8. Notation for the definite integral Pay careful attention to limits: integrating FROM a TO b – see how the a limit is on the bottom of the integral sign?

9. Negative Area If f(x) is below the x-axis, the area “under” the curve (between the function and the x-axis) is negative. http://www.math.rutgers.edu/~greenfie/mill_courses/math135/gifstuff/signed_area.gif

10. Evaluating Integrals Using the Graphing Calculator Later this week, we will learn how to evaluate integrals analytically. For now, we will use the calculator.

11. The function is fnInt and it is in the Math menu

12. The syntax is fnInt(f(x), x, a, b)

13. Example: try the integral of x 2 from 0 to 4

14. FnInt (x^2, x, 0, 4) = 21.33333

15. Comparing Integrals to Riemann Sums Using the Riemann program we installed, calculate the approximate integral of x 2 from 0 to 4 using 4, 8, 16 and 128 sub intervals

16. See that it approaches the correct value: 21.333 as we found out earlier. Subintervals Approximate Sum 4 14 8 17.5 16 19.375 128 21.08

17. Easy Integrals: Geometry The integral of a constant k from a to b is just a rectangle. A=k(b-a)

18. The integral of a linear function is a triangle

19. The integral of a semicircle is A=1/2 p r 2

20. Adding a constant to a function adds a rectangle to the integral: the integral of f(x)+k = the integral of f(x) + k(b-a)