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5.1 Approximating Area Thurs Feb 18 Do Now Evaluate the integral 1)
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Quiz Review Retakes by next Wed
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Sigma Notation Sigma notation: The variable I is called the index of summation. It is the starting value. N is the last value that is plugged into i
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Sigma Notation Examples Find the sum of each
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Summation Theorems If n is any positive integer and c is any constant, then: Sum of constants Sum of n integers Sum of n squares
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Summation Theorems For any constants c and d,
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Examples Ex 2.4 and 2.5 Compute the following sums
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You try Compute the following the sums 1) 2)
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Closure Hand in: Compute the following sum HW: p.297 #27 29 31 35 37 39
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5.1 The Area Problem Fri Feb 19 Do Now Compute each sum
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HW Review: p.297 #27-39 no 33 27) a)45b)24c)99 29) a)-1b)13c)12 31) 15050 35) 1093350 37) 41650 39) -123165
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The Area Problem How do we find the area of certain shapes? We know how to find: –Anything made out of straight lines (squares, triangles, hexagons, etc)
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Estimating the area under a curve We want to find the area under a curve at a certain interval [a, b] 1) Divide the interval into n equal pieces 2) The width of each subinterval is 3) Calculate the height of the curve at each endpoint (plug into f(x)) 4) Find the area of each rectangle formed
![Estimating the area under a curve We want to find the area under a curve at a certain interval [a, b] 1) Divide the interval into n equal pieces 2) The width of each subinterval is 3) Calculate the height of the curve at each endpoint (plug into f(x)) 4) Find the area of each rectangle formed](http://images.slideplayer.com/32/9805918/slides/slide_13.jpg "Estimating the area under a curve We want to find the area under a curve at a certain interval [a, b] 1) Divide the interval into n equal pieces 2) The width of each subinterval is 3) Calculate the height of the curve at each endpoint (plug into f(x)) 4) Find the area of each rectangle formed")
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Endpoints There are 3 types of endpoints (heights) when estimating these sums –Left-endpoint Use the left side of each rectangle –Midpoint Use the average of both endpoints –Right-endpoint Use the right side of each rectangle
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Examples Approximate the area under the curve on the interval [0,1] using 10 right end rectangles
![Examples Approximate the area under the curve on the interval [0,1] using 10 right end rectangles](http://images.slideplayer.com/32/9805918/slides/slide_15.jpg "Examples Approximate the area under the curve on the interval [0,1] using 10 right end rectangles")
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Ex Calculate foron the interval [1,3]
![Ex Calculate foron the interval [1,3]](http://images.slideplayer.com/32/9805918/slides/slide_16.jpg "Ex Calculate foron the interval [1,3]")
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Ex Calculate for the same function [1,3]
![Ex Calculate for the same function [1,3]](http://images.slideplayer.com/32/9805918/slides/slide_17.jpg "Ex Calculate for the same function [1,3]")
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Ex Calculate for on [2,4]
![Ex Calculate for on [2,4]](http://images.slideplayer.com/32/9805918/slides/slide_18.jpg "Ex Calculate for on [2,4]")
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You try: Approximate the area under the curve of y = x^2 on [0,1] using 4 rectangles on a right-endpoint approximation
![You try: Approximate the area under the curve of y = x^2 on [0,1] using 4 rectangles on a right-endpoint approximation](http://images.slideplayer.com/32/9805918/slides/slide_19.jpg "You try: Approximate the area under the curve of y = x^2 on [0,1] using 4 rectangles on a right-endpoint approximation")
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Closure Hand in: Approximate y = x^2 on [0,1] using 4 rectangles with a left-endpoint approximation HW: p.296 #5 7 9 13 15 17
![Closure Hand in: Approximate y = x^2 on [0,1] using 4 rectangles with a left-endpoint approximation HW: p.296 #](http://images.slideplayer.com/32/9805918/slides/slide_20.jpg "Closure Hand in: Approximate y = x^2 on [0,1] using 4 rectangles with a left-endpoint approximation HW: p.296 #")
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5.1 Riemann Sum Practice Thurs Feb 19 Do Now Approximate the area under the curve y = x^2 on [0,5] using 5 rectangles with left-endpoints
![5.1 Riemann Sum Practice Thurs Feb 19 Do Now Approximate the area under the curve y = x^2 on [0,5] using 5 rectangles with left-endpoints](http://images.slideplayer.com/32/9805918/slides/slide_21.jpg "5.1 Riemann Sum Practice Thurs Feb 19 Do Now Approximate the area under the curve y = x^2 on [0,5] using 5 rectangles with left-endpoints")
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HW Review: p. 296 #5 7 9 13 15 17 5) L5 = 46R5 = 44 7) a) L6 = 16.5R6 = 19.5 b) exact area = 18. 9) R3 = 32L3 = 20 13) R3 = 16/3 15) M6 = 87 17) L6 = 12.125
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Practice (green) Worksheet p.349 #3-6, 35-38
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Closure Journal Entry: How do we find the area under a curve? Describe the process. HW: Worksheet p.349 #3-6 35-38
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