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MATH1011 Quizzes

Quiz 12: Uses of integration
Question 1 Questions
An animal population increases at a rate of 1 + 2t per year (where t is measured in years). By how much does the population grow in the first 4 years ?
a)
9
b)
18
c)
20
d)
Cannot be calculated without knowing the initial population

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
04(1 + 2t)dt = t + t2 04 = 4 + 16 0 = 20.
Choice (d) is incorrect
A mosquito population is estimated to be growing at a rate of 2200 + 10e0.8t per week during summer (where t is measured in weeks). By how much does the mosquito population increase between the 4th and the 5th weeks of summer ?
a)
2200 + 10(e4 e3.2)
b)
2200 + 12.5(e4 e3.2)
c)
12.5(e4 e3.2)
d)
Cannot be calculated without knowing the initial population

Choice (a) is incorrect
Choice (b) is correct!
45(2200 + 10e0.8t)dt = 2200t + 12.5e0.8t 45 = 2200 + 12.5(e4 e3.2).
Choice (c) is incorrect
Choice (d) is incorrect
Consider the family of curves sketched below
PIC
The gradient of these curves at the point (t,y) is 8cos2t. What is the general equation of these curves ?
a)
f(t) = 4sin2t + C
 b)
f(t) = 4sin2t + C
c)
f(t) = 16sin2t + C
 d)
f(t) = 16sin2t + C

Choice (a) is correct!
f(t) =8cos2tdt = 8 1 2sin2t + C = 4sin2t + C
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
A family of curves have the slope 8t3 + 9t2 4t + 7 at the point (t,y). Which curve in the family passes through the point (-1,2) ?
a)
f(t) = 24t2 + 18t 4
 b)
f(t) = 2t4 + 3t3 2t2 + 7t + 12
c)
f(t) = 2t4 + 3t3 2t2 + 7t + 2
 d)
f(t) = 24t2 + 18t + 2

Choice (a) is incorrect
Choice (b) is correct!
f(t) =(8t3 + 9t2 4t + 7)dt = 2t4 + 3t3 2t2 + 7t + C
f(1) = 2 3 2 7 + C = 2 C = 12,
therefore,f(t) = 2t4 + 3t3 2t2 + 7t + 12.
Choice (c) is incorrect
Choice (d) is incorrect
Find the area under the curve y = 4 x2 between x = 2 and x = 2.
a)
0
 b)
8 units squared
c)
32 3 units squared
 d)
8 3 units squared

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
22(4 x2)dx = 4x 1 3x3 22
= 8 8 3 8 8 3 = 16 16 3 = 32 3 .
Choice (d) is incorrect
Find the area bounded by the curve y = x3 x2 2x and the x-axis.
a)
9 4 units squared
 b)
37 12 units squared
c)
9 4 units squared
 d)
40 36 units squared

Choice (a) is incorrect
Choice (b) is correct!
PIC
Area = A1 + A2 = 10(x3 x2 2x)dx 02(x3 x2 2x)dx = x4 4 x3 3 x2 10 + x4 4 x3 3 x2 02 = 5 12 8 3 = 37 12.
Choice (c) is incorrect
Choice (d) is incorrect
Find the area between the curves f(x) = cosx and g(x) = sinx between x = 0 and x = π 4 (to two decimal places).
a)
0.29 units squared
 b)
1.71 units squared
c)
0 units squared
 d)
0.41 units squared

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
0π 4 (cosx sinx)dx = sinx + cosx0π 4
= 1 2 + 1 2 (0 + 1) = 2 1 = 0.4142136
Find the area between the curves f(x) = x + 1 and g(x) = x2 x 2.
a)
32 3 units squared
 b)
77 3 units squared
c)
32 3 units squared
 d)
24 units squared

Choice (a) is correct!
The curves intersect at (1,0) and (3,4) so the area is
13(f(x) g(x))dx =13(x2 + 2x + 3)dx = x3 3 + x2 + 3 = 32 3 .
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
After t years from the beginning of 1970 a population is growing at a rate of 2 × 104e0.25t. If the population was 10 000 at the beginning of 1970 find the average population over the period from 1980 to 2005 to 4 significant figures.
a)
5.044 × 107
 b)
8.077 × 107
c)
1.440 × 107
 d)
8.055 × 107

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
P(t) = 104(8e0.25t 7)
therefore,Averagevalue = 1 35 101035104(8e0.25t 7)dt = 400 32(e0.25t 7t) 1035 = 400 32(e8.75 e2.5) 7 × 25 = 8.055 × 107.
Which of the following statements are true?
a)
aa sinxdx = 0 since sinx is an odd function.
b)
aa sinxdx = 20a sinxdx since sinx is an even function.
c)
aax2dx = 0 since x2 is an odd function.
d)
aax2dx = 20ax2dx since x2 is an even function.

There is at least one mistake.
For example, choice (a) should be True.
There is at least one mistake.
For example, choice (b) should be False.
There is at least one mistake.
For example, choice (c) should be False.
There is at least one mistake.
For example, choice (d) should be True.
Correct!
  1. True
  2. False
  3. False
  4. True