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Unformatted text preview: 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 )Δ x 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 → n X i =0 f ( c i )Δ x = Z b a f ( x )d x To help us calculate these areas we have the much-vaunted Fundamental Theorem of Calculus...
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This note was uploaded on 01/07/2010 for the course MAT mat1300 taught by Professor Pieterhofstra during the Fall '09 term at University of Ottawa.
- Fall '09