5.1 Approximating Area Thurs Feb 18 Do Now Evaluate the integral 1)

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Presentation transcript:

5.1 Approximating Area Thurs Feb 18 Do Now Evaluate the integral 1)

Quiz Review Retakes by next Wed

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

Sigma Notation Examples Find the sum of each

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

Summation Theorems For any constants c and d,

Examples Ex 2.4 and 2.5 Compute the following sums

You try Compute the following the sums 1) 2)

Closure Hand in: Compute the following sum HW: p.297 #

5.1 The Area Problem Fri Feb 19 Do Now Compute each sum

HW Review: p.297 #27-39 no 33 27) a)45b)24c)99 29) a)-1b)13c)12 31) ) ) )

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)

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

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

Examples Approximate the area under the curve on the interval [0,1] using 10 right end rectangles

Ex Calculate foron the interval [1,3]

Ex Calculate for the same function [1,3]

Ex Calculate for on [2,4]

You try: Approximate the area under the curve of y = x^2 on [0,1] using 4 rectangles on a right-endpoint approximation

Closure Hand in: Approximate y = x^2 on [0,1] using 4 rectangles with a left-endpoint approximation HW: p.296 #

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

HW Review: p. 296 # ) 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 =

Practice (green) Worksheet p.349 #3-6, 35-38

Closure Journal Entry: How do we find the area under a curve? Describe the process. HW: Worksheet p.349 #