Math 126
Carter
Test 1
Summer 2013
General Instructions: Write your name on only the outside of your blue book. Do all
of your work and write your answers inside your blue book. Put your test paper inside your
blue book as you leave. There are questions on both sides of this page. Solve each of the
following problems. A one-eyed egg sandwich (as in V is for Vendetta) is easy to make.
1. (5 points) State the fundamental theorem of calculus.
2. (10 points) Prove by induction:
N
X
"
N (N + 1)
j3 =
2
j=1
#2
.
3. (8 points each) Compute the following:
(a)
Z
(x2
2x
dx
− 1)4
(b)
Z π/3
sin2 x dx
0
(c)
Z
ex cos (2x)
(d)
Z
cos (3x) sin2 (3x) dx
(e)
Z 5
3x dx
0
(f)
Z ∞
1
1
x101/100
dx
4. (10 points) Compute the total area that lies between the curves y = x3 and y = x for
x ∈ [−1, 1].
5. (10 points) Compute the volume that is obtained by rotating the area that is bounded
by y = 0, x = 1, x = 5 and y = x2 + x about the x-axis.
6. (10 points) Compute
the volume that is obtained by rotating the area under the
√
parabola y = x between x = 1 and x = 3 about the y-axis.
1
7. (10 points) Compute the lateral surface area of the cone that is obtained by rotating
the line y = 10 − x about the x-axis. Recall that the surface area is given by the
integral
Z
b
A=
q
2πf (x) 1 + (f 0 (x))2 dx.
a
8. (10 points)A large container of water originally weighing 144 lbs was lifted at a constant
rate. As it rose, the container leaked at a constant rate. The water was half gone by
the time the container had been lifted to 18 ft. How much work was done lifting the
water? (Neglect the weight of the bag and the lifting equipment.)
2