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Unformatted text preview: 1443 Lecture 2 Mathematics Review Continued Integrals Areas are also very important quantities → How can we calculate the area under a curve? We can approximate this by slicing it up, and add up the area of the slices We will assume that we want the area between 0 and x ∑ = ∆ ≈ N i i x x F A 1 ) ( Now if we let the width of the slices ( ∆ x) get infinitesimally small, we get the true area ∫ ∑ = ∆ = = → ∆ dx x F x x F A N i i x ) ( ) ( lim 1 Now how do we calculate these things? [ ] [ ] [ ] ) ( ) ( ) ( ' ) ( ) ( ' ) ( ) ( ' 1 x F x F d dx x F x F d dx x F dx x F d x F x dx dx = = = = = = ∫ ∫ ∫ ∫ So an integral is the inverse of a derivative… ) ( ) ( ' x F dx x F = ∫ The easiest way to solve them is then to just go backwards…. 1 1 1 + + = ∫ n n x n dx x ) ( 1 ) ( x Sin dx x Cos ϖ ϖ ϖ = ∫ ) ( 1 ) ( x Cos dx x Sin ϖ ϖ ϖ- = ∫ x x e dx e α α α 1 = ∫ ) ln( x x dx = ∫ Now in all of these cases we are assuming that we are getting the area between 0 and x.Now in all of these cases we are assuming that we are getting the area between 0 and x....
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- Fall '08
- Physics, Orders of magnitude, dx