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Quiz 12: Uses of integration

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Question 1

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

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
∫ 4 [ 2]4 0 (1+ 2t) dt = t+ t 0 = 4+ 16- 0 = 20.
Not correct. Choice (d) is false.

Question 2

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)
4 3.2 2200+ 12.5(e - e )
c)
12.5(e4 - e3.2)
d)
Cannot be calculated without knowing the initial population

 

Not correct. Choice (a) is false.
Your answer is correct.
∫ 5 0.8t [ 0.8t]5 4 3.2 (2200+ 10e ) dt = 2200t+ 12.5e 4 = 2200+ 12.5(e - e ). 4
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 3

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

 

Your answer is correct.
∫ ( ) f(t) = 8 cos 2t dt = 8 1sin2t +C = 4sin2t +C 2
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 4

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)
4 3 2 f(t) = 2t + 3t - 2t + 7t+ 2
  d)
f(t) = 24t2 + 18t+ 2

 

Not correct. Choice (a) is false.
Your answer is correct.
∫ f(t) = (8t3 + 9t2 - 4t+ 7) dt = 2t4 + 3t3 - 2t2 + 7t+ C
f(-1) = 2 - 3 - 2 - 7 + C = 2 ⇒ C = 12,
4 3 2 therefore,f(t) = 2t + 3t - 2t +7t +12 .
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 5

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

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
∫ 2 [ ]2 (4- x2) dx = 4x - 1x3 -2 3 -2
( ) ( ) = 8- 8 - - 8- 8 = 16- 16 = 32. 3 3 3 3
Not correct. Choice (d) is false.

Question 6

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

 

Not correct. Choice (a) is false.
Your answer is correct.
PIC
Area = A∫1 0+ A2 ∫ 2 = (x3 - x2 - 2x) dx - (x3 - x2 - 2x) dx -1 0 [x4 x3 ]0 [ x4 x3 ]2 = 4-- -3 - x2 + 4-- -3 - x2 ( ) -1 0 = 5-- - 8 12 3 37 = 12.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 7

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

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
∫ π 4 (cosx - sinx) dx = [sinx + cosx]π4 0 0
- = √1- + 1√--- (0 + 1) = √ 2- 1 = 0.4142136... 2 2

Question 8

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

 

Your answer is correct.
The curves intersect at (-1,0) and (3,4) so the area is
∫ 3 ∫ 3 [ 3 ] (f(x)- g(x)) dx = (- x2 + 2x +3) dx = - x + x2 + 3 = 32. -1 -1 3 3
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.

Question 9

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

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
P (t) = 104(8e0.25t - 7)
1 ∫ 35 therefore,Average value = 35--10- 104(8e0.25t - 7) dt [ 100.25t ]35 = 400 32(e - 7t) 10 = 400 [32(e8.75 - e2.5)- 7 ×25] 7 = 8.055× 10 .

Question 10

Which of the following statements are true?

a)
∫ a sinx dx = 0 -a since sinx is an odd function.
b)
∫ a ∫ a -a sinx dx = 2 0 sin x dx since sinx is an even function.
c)
∫ a 2 -a x dx = 0 since x2 is an odd function.
d)
∫ ∫ a 2 a 2 -a x dx = 2 0 x dx 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.
Your answers are correct
  1. True.
  2. False.
  3. False.
  4. True.
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