Workshop_Solutions_5

Workshop_Solutions_5 - Mathematical Methods for Physics 1...

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Mathematical Methods for Physics 1, Solutions to Workshop Sheet 5 1 Mathematical Methods for Physics 1, Autumn 2012 – Department of Mathematics, University of Sussex Solutions to Workshop Sheet 5 1. Find the area under the graph of the function f ( x ) = 1 x , where x n = 0 , between (a) x = 1 and x = 2, (b) x = 2 and x = 1. (c) Sketch the graph of f ( x ) = 1 /x and indicate the areas under the graph from (a) and (b). What is the relation between the answer from (a) and (b)? Explain why this can be seen from your sketch. Solution: (a) The areas is i 2 1 f ( x ) dx = i 2 1 1 x dx = ( ln | x | )v v 2 1 = ln(2) ln(1) = ln(2) . (b) The area is i 1 2 f ( x ) dx = i 1 2 1 x dx = ( ln | x | )v v 1 2 = ln(1) ln(2) = ln(2) . (c) We sketch the graph of f ( x ) = 1 /x and indicate the areas from (a) and (b). 2 1 1 2 -1 -2 -1 -2 Figure 1: Graph of f ( x ) = 1 /x and the area from x = 1 to x = 2 and the area from x = 2 to x = 1. We observe that i 2 1 1 x dx = ln(2) = i 1 2 1 x dx.

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2 Mathematical Methods for Physics 1, Solutions to Workshop Sheet 5 This is to be expected from the graph of the odd function f ( x ) = 1 /x for the following reasons: Once we have indicated the areas under the graph from x = 1 to x = 2 and from x = 2 to x = 1, we see that both areas are the same (because f ( x ) = 1 /x is odd, see also Problem 2),
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Workshop_Solutions_5 - Mathematical Methods for Physics 1...

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