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Summation Notation Also called sigma notationAlso 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: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 5 th term like before.

Examples: Write each series in summation notation. 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:

Example: Find the sum of the series. k goes from 5 to 10.k goes from 5 to 10. (5 2 +1)+(6 2 +1)+(7 2 +1)+(8 2 +1)+(9 2 +1)+(10 2 +1)(5 2 +1)+(6 2 +1)+(7 2 +1)+(8 2 +1)+(9 2 +1)+(10 2 +1) = 26+37+50+65+82+101 = 26+37+50+65+82+101 = 361

You try some. Find the Sum.

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

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

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. This is the Midpoint Rectangular Approximation Method (MRAM). Approximate area: In this example there are four subintervals. As the number of subintervals increases, so does the accuracy.

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

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

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

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

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

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

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!

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

min f max f 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. h

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

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.)

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

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

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 and Areas 01 It can be used to find an area bounded, in part, by a curve e.g. gives the area shaded on the graph The limits of integration... Definite integration results in a value. Areas

Definite Integration and Areas... give the boundaries of the area. The limits of integration... 01 It can be used to find an area bounded, in part, by a curve Definite integration results in a value. Areas x = 0 is the lower limit ( the left hand boundary ) x = 1 is the upper limit (the right hand boundary ) 0 1 e.g. gives the area shaded on the graph

Definite Integration and Areas... give the boundaries of the area. The limits of integration... 01 It can be used to find an area bounded, in part, by a curve Definite integration results in a value. Areas x = 0 is the lower limit ( the left hand boundary ) x = 1 is the upper limit (the right hand boundary ) 0 1 e.g. gives the area shaded on the graph

Definite Integration and Areas 01 23 2  xy Finding an area the shaded area equals 3 The units are usually unknown in this type of question Since

Definite Integration and Areas “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

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