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4.1 Areas and Distances

# 4.1 Areas and Distances - The limits as n of both the lower...

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Areas and Distance Section 4.1

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Sigma Notation The sum of n terms a1, a2, a3,….,an is written as Where i is the index of summation , ai is the i th term of the sum, and the upper and lower bounds of summation are n and 1. n n i i a a a a a ..... 3 2 1 1 + + + = =
Summation Formulas 1. 2. 3. = = n i cn c 1 = + = n i n n i 1 2 ) 1 ( 6 ) 1 2 )( 1 ( 1 2 + + = = n n n i n i

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4. 4 ) 1 ( 2 2 1 3 + = = n n i n i
Area of a Plane Region The area of a parabolic region is greater than area of the rectangles. The area of the parabolic region is less than the area of the rectangles.

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Upper and Lower Sums To approximate the area of the region, begin by subdividing the interval [a, b] into n subintervals, each of length a b f(mi) f(Mi) n a b x - = x
Lower sum = Area of inscribed rectangles Upper sum = Area of circumscribed rectangles x m f n s n i i = = ) ( ) ( 1 x m f n s n i i = = ) ( ) ( 1 x M f n S n i i = = ) ( ) ( 1

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Left endpoints: Right endpoints : x i a - + ) 1 ( x i a +
Limit of the Lower & the Upper Sums Let f be continuous and nonnegative on the interval [a, b].

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Unformatted text preview: The limits as n of both the lower and upper sums exist and are equal to each other. That is x m f n s n i i n n ∆ = ∑ = ∞ → ∞ → ) ( lim ) ( lim 1 Where f (mi) and f (Mi) are the minimum and maximum values of f on the i th subinterval. x M f n i i n ∆ = ∑ = ∞ → ) ( lim 1 ) ( lim n S n ∞ → = Defn. of the Area of a Region in the Plane Let f be continuous and nonnegative on the interval [a, b]. The area of the region bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b is x c f Area n i i n ∆ = ∑ = ∞ → ) ( lim 1 Distance 1 →∞ = ∆ ∑ n i ( ) lim f t t Joke Time What is 7Q + 3Q? You’re welcome! How do hearing impaired people greet one another? They sine waves...
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4.1 Areas and Distances - The limits as n of both the lower...

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