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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
xaxis.
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 nonconstant 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 xaxis.
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 viceversa. ...
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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.
 Spring '11
 Blake
 Math, Calculus

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