MATH 101 HOMEWORK 7 – SOLUTIONS
1. The speedometer reading (v) on a car was observed through a 10-minute period at
1-minute intervals and recorded in the table below. Use Simpson’s Rule to estimate the
distance travelled by the car. (The total distance travelled is the integral of the velocity.
All readings are in kilometers per hour. In this problem, you may use a calculator.)
t
v
S10 =
0
56
1
58
2
62
3
61
4
64
5
66
6
63
7
60
8
61
9
63
10
61
1
(56 + 4 · 58 + 2 · 62 + 4 · 61 + 2 · 64 + 4 · 66 + 2 · 63 + 4 · 60 + 2 · 61 + 4 · 63 + 61)
3 · 60
= 10.27222 . . . km.
We divided v by 60 to get the velocity in kilometers per minute.
Z
∞
dx
,
1 + x6
1
but we can’t do it directly because the interval of integration is infinite. Use a substitution
to convert this integral to one over a finite interval (see the examples on page 398 of the
textbook).
2. We want to use a numerical method (eg. Simpson’s Rule) to evaluate
√
We substitute t = 1/x, dx = −dt/t2 :
Z ∞
Z 1
Z 1
Z 1
dx
dt
dt
tdt
√
√
√
√
=
=
=
,
2 1 + t−6
2 −3 t6 + 1
1 + x6
t6 + 1
1
0 t
0 t t
0
which is a proper integral that can be evaluated by numerical methods.
3. Calculate the volume of each solid, using either slices or cylindrical shells, whichever
you prefer:
(a) The solid generated by rotating about the x-axis the region bounded by the curves
y = cos x, y = sin x, and the lines x = 0, x = π/4.
Use slices:
Z
Z
π/4
π/4
(cos x − sin x)dx = π
2
π
0
2
cos(2x)dx =
0
1
π/4 π
π
sin(2x)
= .
2
2
0
(b) The solid generated by rotating about the y-axis the region bounded by the curve y = ex ,
and the lines x = 1, y = 1.
Use cylindrical shells:
Z
Z
1
2πx(e − 1)dx =
0
Z
1
1
2πxe dx −
x
0
1
x
Z
2πxdx = 2π(xe −
x
0
0
1
x
0
= 2πe − 2πe − π = 2πe − 2πe + 2π − π = π.
0
2
1
1
ex dx) − πx2 0