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homework3 - (a Let y = sin x and 0 ≤ x ≤ π Revolve the...

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M408C Homework: 10/30 - 11/20 10/30 No homework assigned. 11/1 From the textbook: p.493, #1, 3, 5, 7, 15, 20. Show that sec x dx = ln | sec x + tan x | by di ff erentiating. Integrate the following by parts, and check that the result is correct by di ff erentiating. ( a ) x sin x dx ( b ) x 2 e x dx 11/6 No homework assigned. 11/8 Integrate the following: (a) e x sin x dx (b) x 5 1 + x 3 dx (c) x 1 + x 2 dx (d) x 3 1 + x 2 dx (e) e x x dx (f) sin 3 x cos x dx (g) x + 1 2 x 2 + 4 x + 1 dx (h) x 1 + x 4 dx (i) sec 2 x 1 - tan 2 x dx 11/13 Integrate the following: (a) (3 x + 2)(9 x - 3) 3 / 2 dx (b) x 5 ( x 3 + 1) 3 / 2 dx (c) 3 x 2 tan 4 ( x 3 ) sec 2 ( x 3 ) dx (d) cos 3 x sin x x dx (e) x 3 e sin( x 4 ) cos( x 4 ) dx
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11/15 Given velocity v , find both displacement and total distance traveled. (a) v ( t ) = t 2 - 4 t + 3, where 0 < t < 5 (b) v ( t ) = t ( t + 1)( t - 2) = t 3 - t 2 - 2 t , where - 1 < t < 4 Let B ( t ) be the amount of rain in a barrel. Suppose B ( t ) = t 2 + t in/hr of rain. (c) Find total accumulation for 2 < t < 4. (d) If B (3) = 20, find B ( t ). 11/20 Find the volumes of the solids obtained by revolving the indicated regions about the x-axis.
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Unformatted text preview: (a) Let y = sin x and 0 ≤ x ≤ π . Revolve the region under the curve and above the x-axis. [Note that sin 2 x = 1 / 2-cos(2 x ) / 2.] (b) Let y 1 = 4-x 2 and y 2 = x 2 . Revolve the region between y 1 , y 2 , and the y-axis. (c) Revolve the region enclosed by the circle of radius 1 centered at (0 , 3). Here, just set up an integral that describes the volume of the resulting torus. Also, (d) Find the volume of a right circular cone of height H and base radius R. (e) Find the volume of a solid object with a circular base and square cross sections. Finally, if y 1 = 4-x 2 and y 2 = x 2 , find the area of following regions. (f) The region between y 1 , y 2 , and the y-axis, (g) The region between y 1 , y 2 , and the x-axis....
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