MIT18_014F10_pr_ex1_sols

MIT18_014F10_pr_ex1_sols - Practice Exam 1- Solutions...

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Unformatted text preview: Practice Exam 1- Solutions September 29, 2010 Problem 1: Compute 103 (2 x 198) 2 x 99 dx where here [ x ] is defined 99 to be the largest integer x . Solution By the properties of the integral, we know that the above is equal to 4 4 ( x ) 2 x dx. Now, [ x ] takes the value on [0 , 1) and the value 1 on [1 , 4) and the value 2 at x = 4. Thus, we can rewrite the integral 4 4 4 3 1 3 4 x 2 dx + 4 x 2 dx = 4 3 4 = 256 / 3 4 / 3 = 252 / 3 = 84 . 3 1 4 Problem 2: Let S be a square pyramid with base area r 2 and height h . Using Cavalieris Theorem, determine the volume of the pyramid. Solution Orient the square pyramid so that the base sits on the x y plane and the top vertex sits on the z axis. Let a S ( h ) denote the cross-sectional area of S { z = h } , and note that this is a square that will be a function of r,h . To find the length of a side of the square at height h , we consider the line which contains the two points ( r, 0) and (0 ,h ). One form of the equation for this line is y = r h x + h . Notice that here x is the side length of the square at...
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MIT18_014F10_pr_ex1_sols - Practice Exam 1- Solutions...

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