finpracB - Part 1: Integration problems from 2004-05 exams...

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Part 1: Integration problems from 2004-05 exams 1. Find each of the following. (a) Z ± 4 t 4 - t - 1 + 2 t 2 + 1 ² dt (b) Z ± sec x tan x - sin x + x 2 + 1 3 x ² dx (a 0 ) Z 2 0 x 2 e x 3 dx (b 0 ) Z 1 0 x 2 e 3 x dx 2. (a) Let f ( x ) = e x . On the graph of f pictured below, draw the approximating rectangles that are used to estimate the area under the curve between x = 0 and x = 1 according to the the Left Endpoint Rule; use n = 4 rectangles. 1 1 2 3 (b) Write an expression involving only numbers (including e ) that represents the area esti- mate using these rectangles. Is your quantity an underestimate or an overestimate of the actual area? (c) Write a mathematical statement that expresses the area under the curve from x = 0 to x = 1 as a limit, again using the Left Endpoint Rule. Explain any notation you use. (You should not evaluate this limit.) (d) Calculate the value of the definite integral Z 1 0 e x dx , simplifying your answer as much as you can. 3. Evaluate the following integrals, showing all of your work. (a) Z x 3 1 - x 2 dx (c) Z x 2 (ln x ) 2 dx
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4. A particle moves along a line, with acceleration (in meters/sec 2 ) as a function of time t (in seconds) given by a ( t ) = 2 t - 7 . Furthermore, at time t = 1 second, the particle’s velocity is 4 meters per second. (a) Find the particle’s velocity function v ( t ). (b) What is the particle’s net change in position (i.e., its displacement) between the times
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This note was uploaded on 01/12/2010 for the course MATH 41 at Stanford.

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finpracB - Part 1: Integration problems from 2004-05 exams...

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