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.

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.

Examples: Write each series in summation notation. 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:

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

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

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. time velocity After 4 seconds, the object has gone 12 feet.

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). Approximate area:

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

Another approach would be to use rectangles that touch at the midpoint 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. Approximate area:

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

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. subinterval If we let n = number of subintervals, then

Leibnitz introduced a simpler notation for the definite integral:

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

Since rate . time = distance: 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. time velocity After 4 seconds, the object has gone 12 feet.

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.

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.

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

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

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

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

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

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

x = 0 is the lower limit x = 1 is the upper limit Areas Definite integration results in a value. It can be used to find an area bounded, in part, by a curve 1 e.g. gives the area shaded on the graph 1 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 )

x = 0 is the lower limit x = 1 is the upper limit Areas Definite integration results in a value. It can be used to find an area bounded, in part, by a curve 1 e.g. gives the area shaded on the graph 1 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 )

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

Rules for Definite Integrals Order of Integration:

Rules for Definite Integrals Zero:

Rules for Definite Integrals 3) Constant Multiple:

Rules for Definite Integrals 4) Sum and Difference:

Rules for Definite Integrals 5) Additivity:

Using the rules for integration Suppose: Find each of the following integrals, if possible: b) c) d) e) f)

“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