Summation Notation

1. Summation Notation • Also called sigma notation (sigma is a Greek letter Σ meaning “sum”) The series 2+4+6+8+10 can be written as: i is called the index of summation Sometimes you will see an n or k here instead of i. The notation is read: “the sum from i=1 to 5 of 2i” i goes from 1 to 5.

2. Summation Notation for an Infinite Series • Summation notation for the infinite series: 2+4+6+8+10+… would be written as: Because the series is infinite, you must use i from 1 to infinity (∞) instead of stopping at the 5th term like before.

3. a. 4+8+12+…+100 Notice the series can be written as: 4(1)+4(2)+4(3)+…+4(25) Or 4(i) where i goes from 1 to 25. Notice the series can be written as: Examples: Write each series in summation notation.

4. Example: Find the sum of the series. • k goes from 5 to 10. • (52+1)+(62+1)+(72+1)+(82+1)+(92+1)+(102+1) = 26+37+50+65+82+101 = 361

5. You try some. Find the Sum.

6. Photo by Vickie Kelly, 2002 Greg Kelly, Hanford High School, Richland, Washington Estimating with Finite Sums Estimating with Finite Sums Greenfield Village, Michigan

7. velocity time Consider an object moving at a constant rate of 3 ft/sec. Since rate . time = distance: If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line. After 4 seconds, the object has gone 12 feet.

8. Approximate area: If the velocity is not constant, we might guess that the distance traveled is still equal to the area under the curve. (The units work out.) Example: We could estimate the area under the curve by drawing rectangles touching at their left corners. This is called the Left-hand Rectangular Approximation Method (LRAM).

9. Approximate area: We could also use a Right-hand Rectangular Approximation Method (RRAM).

10. Approximate area: Another approach would be to use rectangles that touch at the midpoint. This is the Midpoint Rectangular Approximation Method (MRAM). In this example there are four subintervals. As the number of subintervals increases, so does the accuracy.

11. Approximate area: The exact answer for this problem is . With 8 subintervals: width of subinterval

12. Circumscribed rectangles are all above the curve: Inscribed rectangles are all below the curve:

13. When we find the area under a curve by adding rectangles, the answer is called a Rieman sum. The width of a rectangle is called a subinterval. The entire interval is called the partition. subinterval partition If we let n = number of subintervals, then

14. Leibnitz introduced a simpler notation for the definite integral: Note that the very small change in x becomes dx.

15. It is called a dummy variable because the answer does not depend on the variable chosen. upper limit of integration Integration Symbol integrand variable of integration (dummy variable) lower limit of integration

16. velocity time Earlier, we considered an object moving at a constant rate of 3 ft/sec. Since rate . time = distance: If we draw a graph of the velocity, the distance that the object travels is equal to the area under the line. After 4 seconds, the object has gone 12 feet.

17. If the velocity varies: Distance: (C=0 since s=0 at t=0) After 4 seconds: The distance is still equal to the area under the curve! Notice that the area is a trapezoid.

18. What if: We could split the area under the curve into a lot of thin trapezoids, and each trapezoid would behave like the large one in the previous example. It seems reasonable that the distance will equal the area under the curve.

19. The area under the curve We can use anti-derivatives to find the area under a curve!

20. Let area under the curve from a to x. (“a” is a constant) Let’s look at it another way: Then:

21. min f max f h The area of a rectangle drawn under the curve would be less than the actual area under the curve. The area of a rectangle drawn above the curve would be more than the actual area under the curve.

22. As h gets smaller, min f and max f get closer together. This is the definition of derivative! initial value Take the anti-derivative of both sides to find an explicit formula for area.

23. As h gets smaller, min f and max f get closer together. (Area under curve from a to x ) = (antiderivative at x minus antiderivative at a.)

24. Fundamental Theorem of Calculus Area “The essence of mathematics is not to make simple things complicated, but to make complicated things simple.” -- Gudder

25. Example: Find the area under the curve from x=1 to x=2. Area under the curve from x=1 to x=2. Area from x=0 to x=2 Area from x=0 to x=1

26. The constant always cancels when finding a definite integral, so we leave it out! Integrals such as are called definite integrals because we can find a definite value for the answer.

27. When finding indefinite integrals, we always include the “plus C”. Integrals such as are called indefinite integrals because we can not find a definite value for the answer.

28. Definite integration results in a value. e.g. gives the area shaded on the graph 0 1 Areas It can be used to find an area bounded, in part, by a curve The limits of integration . ..

29. Definite integration results in a value. 1 e.g. gives the area shaded on the graph 0 0 1 Areas It can be used to find an area bounded, in part, by a curve The limits of integration . . . . . . give the boundaries of the area. x = 0 is the lower limit ( the left hand boundary ) x= 1 is the upper limit (the right hand boundary )

30. Definite integration results in a value. 1 e.g. gives the area shaded on the graph 0 0 1 Areas It can be used to find an area bounded, in part, by a curve The limits of integration . . . . . . give the boundaries of the area. x = 0 is the lower limit ( the left hand boundary ) x= 1 is the upper limit (the right hand boundary )

31. 2 = + y 3 x 2 Since 0 1 Finding an area the shaded area equals 3 The units are usually unknown in this type of question

32. “Thus mathematics may be defined as the subject in which we never know what we are talking about, not whether what we are saying is true.” -- Russell