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http://ocw.mit.edu 18.01 Single Variable Calculus
Problem 1. Problem 2. Evaluate � Evaluate � x−4
(x + 1)(x2 + 4)
0 (x2 dx
by making the substitution x = 2 tan u.
+ 4)2 Problem 3.
a) Derive a reduction formula relating
b) Let F (x) =
e−x dx. Express
0 � 1
2n −x2 xe
0 dx to � 1
2 x2n−2 e−x dx.
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.
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) Compute the arclength of the curve.
b) Compute the surface area of the surface formed by rotating the curve around the
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.
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.
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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