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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.

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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.

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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:

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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

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You try some. Find the Sum.

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

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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.

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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:

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

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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.

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

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

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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

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

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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.

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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.

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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.

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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.

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

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

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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

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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

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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.)

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

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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.

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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!

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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”.

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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

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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

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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

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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

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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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