Engineering Mathematics One (ENGE501):
Differential and Integral Calculus (MATH501)
Assignment Four
Due Monday 29 May 2017
Please ensure that all your working is clearly set out, all pages are stapled
together, and that your name along with the course name, your lecturer
name and your stream number (ex. ENGE501/10 etc. or MATH501/10)
appears on the front page of the assignment. Please place completed assignments
in the assignment box in WT level 1 by 4:30pm on the due date. Name
Student ID
Course name
Paper Code/Stream
Lecturer’s Name Questions:
1. Evaluate each of the following definite integrals:
R4 R15
0 1
√
3 x R1 dx

2e3x − 4 cos 2x dx 1
7−2t dt
1
0 (7−2t)2 dt R01 2. Consider the square with vertices (0,0), (0,1), (1,1) and (1,0). The parabola
y = x2 splits the square into two unequal parts. Find the ratio of the
smaller area to the larger one.
3. Evaluate each of the following indefinite integrals:
R


R 4


x 1 + 3x5 dx
x 1 + 3x5 dx
R cos θ
R 5x2 −4x+7
(x2 +1)(2x−1) dx
sin2 θ dθ
4. Evaluate the improper integral R1 5x2 −4x+7
0 (x2 +1)(2x−1) 1 dx. 5. The force acting on a particle on the x-axis is
F (x) = 9800
,
(x + 500)2 when x≥0 (a) Find the work required to move the particle from x = 0 to x = 20.
(b) By evaluating an improper integral, find the amount of work required
to move the particle from x = 0 to x = ∞
6. A heavy chain that weighs 8 kilograms/metre is initially hanging 20 metres
down into a mine shaft. How much work is done in pulling up the chain
to the top of the mine shaft? (You can take g = 9.81ms−2 ).
7. The rate at which a pollutant is being dumped into a small stream at time
t is given by P0 e−kt , where P0 is the rate of the pollutant initially released
into the stream. Suppose P0 = 1000 and k = 0.06. Find the total amount
of the pollutant that will be released into the stream into the indefinite
future.
8. When mixing a certain cocktail a barperson starts with a 5 litre solution
of 10% alcohol. Into this solution the barperson mixes vodka (40% alcohol) at the rate of 2 litres/minute and pours into glasses at the rate of 3
litres/minute.
(a) Find a formula for the volume V (t) of the mixture at time t minutes.
(b) Find a differential equation that tells the rate of change of the amount
of alcohol A(t) at time t minutes.
(c) Solve the differential equation and use the initial condition to find a
formula for A(t)
9. When a charged capacitor is connected to earth, it discharges and the
voltage V across it will decrease according to the equation:
−V
dV
=
dt
RC
where R (the resistance) and C (the capacitance) are constants. If the
voltage V across a certain capacitor with capacitance C = 3 × 10−6 farads
drops from 10 volts to 1 volt in two seconds, determine the resistance R. 2

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