Charles Darwin University
SMA102- Mathematics 1B
Assignment 1
Notes: Please submit your assignment solutions as a single pdf file in Learnline. Show
all workings in each question clearly and in detail, else you will lose marks.
1. Evaluate the following indefinite integrals:
(i)
∫ ( ln x )2 dx
(ii)
1
∫ csc 2 x −cot 2 x dx
[15]
2. Evaluate the following definite integrals:
π /4
1
dx
∫ 2+ tan
x
0
[15]
3. (i) Using the reduction formula, show that if I n=∫ x cos x dx , then
n
∫ x n cos x dx=¿ x n sin x +nx n−1 cos x−n(n−1) I n−2 ¿
[15]
(ii) Hence evaluate
π
∫ x 5 cos x dx
0
4. Find the area of the shaded region by
[5]
(i)
integrating with respect to x
(ii)
integrating with respect to y
[15]
5. Use cylindrical shells to find the volume of the solid that is generated when the region
that is enclosed by y=x 3, y=1 , x=0 is revolved about the line y = 1.
First sketch the cylinder for the volume of the solid and show the height and radius of
the cylinder.
[15]
6.
i.
Sketch one loop of the curve
2
2
9 y =x (3−x)
ii.
Find the area of the surface of revolution generated by one loop of the curve
2
2
9 y =x (3−x) about the y-axis. Leave your answer in the exact form.
[20]
Marking Scheme
The following table forms the basis for the marking of assignment 1. Show all workings in
detail with explanation, else you will lose marks.
Questi
on
1)
Section description
Mark
s
Evaluate the following indefinite integrals:
7
∫ ( ln x )2 dx
i)
1
∫ csc 2 x −cot 2 x dx
Use the correct integration method and show detailed working.
2)
Evaluate the following definite integrals:
π /4
1
dx
∫ 2+ tan
x
15
0
3)
i)
8
Using the reduction formula, show that if I n=∫ x cos x dx , then
∫ x n cos x dx=¿ x n sin x +nx n−1 cos x−n(n−1) I n−2 ¿
n
15
[5
Hence use the solution in part i) to evaluate
π
∫ x 5 cos x dx
ii)
0
5
4)
i)
integrating with respect to x
7
ii)
integrating with respect to y
8
Sketch to show the subareas in the integrand or explain the terms in your
integral.
5)
6)
Sketch the cylinder for the volume of the solid that is generated when the
region that is enclosed by y=x 3, y=1 , x=0 is revolved about the line y = 1.
Show the height and radius of the cylinder.
3
Write an expression for the volume of the cylindrical shell
2
Evaluate the volume
10
i)
Sketch one loop of the curve
ii)
Find the area of the surface of revolution generated by one loop of the curve
2
2
15
9 y =x (3−x) about the y-axis and leave your answer in the exact form.
Total
2
2
9 y =x (3−x)
5
100