MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Fall 2006� � � � � � 18.01 Practice Questions for Exam 4 – Fall 2006 x − 4 Problem 1. Evaluate dx. (x + 1)(x2 + 4) 2 dx Problem 2. Evaluate by making the substitution x = 2 tan u. (x2 + 4)2 0 Problem 3. 1 1 2 2 a) Derive a reduction formula relating x 2n e −x dx to x 2n−2 e −x dx. 0 0 1 1 b) Let F (x) = e −x 2 dx. Express x 2 e −x 2 dx in terms of values of F (x). 0 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 � = r2, 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
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