2311lectureoutline5.1

# 2311lectureoutline5.1 - the height is determined by the...

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MAC2311, Calculus I 5.1 Areas To approximate the area under a curve, we can approximate the region by using rectangles and letting the number of rectangles become large. The precise area is the limit of these sums of areas of rectangles. A right sum, R n , is the sum of the area of rectangles under the curve f(x) where x Δ is the width of each rectangle and f(x i ) is the height of each rectangle, where the height is determined by the right side of the rectangle touching the curve. A left sum, L n , is the sum of the area of rectangles under the curve f(x) where x Δ is the width of each rectangle and f(x i ) is the height of each rectangle, where
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Unformatted text preview: the height is determined by the left side of the rectangle touching the curve. Definition: Let a function f be defined on the interval [a, b] and let f(x)> 0 for all x in [a,b]. Then the area A of the region that lies between the graph of f and the x-axis is the limit of the sum of the areas of approximating rectangles: 1 2 1 [ ( ) ( ) ... ( ) ] lim lim lim ( ) n n n i i n n n i A R f x x f x x f x x f x x →∞ →∞ →∞ = = = Δ + Δ + + Δ = Δ & where b a x n-Δ = and i x a i x = + Δ , provided the limit exists. Try exercises 2, 4, 18, 24 on pages 364-365 in your text....
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