lecture15 - Lecture 15 Areas and the Denite Integral 15.1...

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Lecture 15 - Areas and the Definite Integral 15.1 Areas and the Definite Integral Suppose that f is non-negative on a closed interval [ a, b ]. We wish to calculate the area underneath the curve but below the x -axis between a and b : One approach is to use approximating rectangles: The areas of rectangles are easy to calculate. If each width is Δ x = b - a n where n is the number of rectangles, then the total area is n X i =0 f ( c i x
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The actual area is obtained by taking the limit as n → ∞ , or equivalently, as Δ x 0. The problem is that in most cases this limit is hard to calculate directly, even for fairly simple functions. Thankfully, there is a very powerful theorem that helps us calculate this easily. Extremely Suggestive Notation: We call the limit above the definite integral and denote it R b a f ( x ) d x . In other words, lim Δ x 0 n X i =0 f ( c i x = Z b a f ( x ) d x To help us calculate these areas we have the much-vaunted Fundamental Theorem of Calculus Theorem 15.1 (The Fundamental Theorem of Calculus):
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