The University of Sydney
School of Mathematics and Statistics
Tutorial for Week 13
MATH1011: Applications of Calculus
Semester 1, 2020
Web Page:
www.maths.usyd.edu.au/MATH1011
Lecturer: Clio Cresswell, Brad Roberts
Tutorial exercises:
1.
Find the following indefinite integrals using substitution or recognition.
(a)
Z
t
e

t
2
d
t
.
(b)
Z
6
s
cos(3
s
2
+ 2) d
s
.
(c)
Z
tan
x dx.
Hint:
Write tan
x
=
sin
x
cos
x
and choose a suitable substitution.
2.
After
t
years from the beginning of 1990, a population is growing at the rate of 4
×
10
5
e
0
.
2
t
.
If the population was 2
×
10
6
at the beginning of 1990 (i.e. at
t
= 0), find the average
population over the period from the beginning of 1990 to the beginning of 2004.
3.
The function
T
(
t
) = 37 sin
2
π
365
(
t

101)
+ 25 was used as an approximation to the
temperature (in degrees Fahrenheit) in Fairbanks, Alaska, over a 365day year (
t
in
days). Use this function to find the average temperature in Fairbanks
(a)
for that year,
(b)
for the first 100 days of that year.
4.
Find the area enclosed by the curves
y
=
x
2

4
x

5 and
y
= 3

2
x
.
5.
Let
S
be the region bounded by the curves
y
=
x
4

x
2
and
y
= 1

x
2
.
(a)
Draw a neat sketch of
S
.
(b)
Use definite integrals to calculate the area of
S
.
6.
It has been computed that the wait for a car parking space in a parking lot is given by
f
(
t
) = 0
.
1
e

0
.
1
t
where
t
is the number of minutes waiting and so
t
≥
0.
(a)
Verify that this is in fact a probability density function.
(b)
Determine the probability that a person will wait for a parking space for over 6
minutes.
7.
Evaluate the following improper integrals (if possible):
(a)
Z
∞
1
d
x
x
5
/
2
(b)
Z
∞
3
d
x
x

2