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workshop5

# workshop5 - R around the y-axis 3 Suppose f x = x-ln x a...

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Workshop 5 1) a) Compute R 2 1 dx x 2 . b) Compute R 2 1 dx x ( x - m ) if m is a small positive number. What happens when m 0 + ? c) Compute R 2 1 1 x 2 + n dx if n is a small positive number. What happens when n 0 + ? d) Sketch a graph of 1 x 2 , 1 x ( x - m ) , and 1 x 2 + n if m and n are both . 1 for x between 1 and 2. 2) Suppose that a is a positive constant and that R is the region bounded above by y = 1 /x a , below by y = 0, and on the left by the line x = 1 . a) Sketch the curves y = 1 /x a for a = . 5, 1 and 2. Which of these is closest to the x -axis? b) For which positive numbers a do you get a convergent integral when you attempt to calculate the area of R ? c) Same as b), but for the volume of the solid obtained by rotating R around the x -axis. d) Same as c), but for the volume of the solid obtained by rotating
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Unformatted text preview: R around the y-axis. 3) Suppose f ( x ) = x-ln x . a) Verify that lim x → + f ( x ) = 0 and lim x →∞ f ( x ) = 0. Graph f on the interval [0 , 10]. b) A remarkable result of third semester calculus is that R ∞-∞ e-x 2 dx = √ π . Assume that this result is correct, and use it to show that R ∞ x-ln x dx = e 1 / 4 √ π . ( Maple can “do” the ﬁrst integral, but not the second!) Hint Make a change of variables, and then complete the square. c) Include a graph of e-x 2 on the interval [-2 , 2]. Use your answer to b) to compare this graph to the graph in a)....
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