Practice Questions

# Practice Questions - MIT OpenCourseWare http://ocw.mit.edu...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.01 Practice Questions for Exam 4 – Fall 2006 Problem 1. Problem 2. Evaluate � Evaluate � x−4 dx. (x + 1)(x2 + 4) 2 0 (x2 dx by making the substitution x = 2 tan u. + 4)2 Problem 3. a) Derive a reduction formula relating �1 � 2 b) Let F (x) = e−x dx. Express 0 � 1 2n −x2 xe 0 dx to � 1 2 x2n−2 e−x dx. 0 1 2 −x2 xe dx in terms of values of F (x). 0 Problem 4. Find the volume of the solid obtained by rotating about the y -axis the ﬁnite region bounded by the positive x- and y -axes and the graph of y = cos x. Problem 5. Make a reasonable sketch of one loop of the polar curve r = sin 3�, and ﬁnd the area inside it. Problem 6. Let x(t) = cos3 t, y (t) = sin3 t, 0 � t � � /2 be a parametric representation of a curve. a) Compute the arclength of the curve. b) Compute the surface area of the surface formed by rotating the curve around the x-axis. Problem 7. Set up an integral for the length of one arch of the curve y = sin x, and by � estimating the integral, tell how this length compares with � 2. Problem 8. A circular metal disc of radius a has a non-constant density � (units: gms/cm2 ); the density at a point P on the disc is given by � = r 2 , where r is the distance of the point from the center of the disc. Set up and evaluate a deﬁnite integral giving the total mass of the disc. Problem 9. a) Sketch the curve given in polar coordinates by r = 1 + cos � b) Find the polar coordinates of the following two points (show work): (i) where the curve in part (a) intersects the circle of radius 3/2 centered at the origin; (ii) where the above curve intersects the circle of radius 3/2 centered at the point x = 3/2 on the x-axis. Other kinds of problems: Other kinds of partial fractions decompositions; sketching curves given parametrically, ﬁnding their arclength; ﬁnding surface area for rotated curves in xy -coordinates; deriving polar equations of curves given geometrically, changing from rectangular equa­ tions to polar and vice-versa. ...
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## This note was uploaded on 10/10/2011 for the course MATH 31 taught by Professor Blake during the Spring '11 term at MIT.

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Practice Questions - MIT OpenCourseWare http://ocw.mit.edu...

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