 ## H1011 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!
${\int }_{0}^{4}\left(1+2t\right)\phantom{\rule{1em}{0ex}}dt={\left[t+{t}^{2}\right]}_{0}^{4}=4+16-0=20.$
Choice (d) is incorrect
A mosquito population is estimated to be growing at a rate of $2200+10{e}^{0.8t}$ 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) $2200+10\left({e}^{4}-{e}^{3.2}\right)$ b) $2200+12.5\left({e}^{4}-{e}^{3.2}\right)$ c) $12.5\left({e}^{4}-{e}^{3.2}\right)$ d) Cannot be calculated without knowing the initial population

Choice (a) is incorrect
Choice (b) is correct!
${\int }_{4}^{5}\left(2200+10{e}^{0.8t}\right)\phantom{\rule{1em}{0ex}}dt={\left[2200t+12.5{e}^{0.8t}\right]}_{4}^{5}=2200+12.5\left({e}^{4}-{e}^{3.2}\right).$
Choice (c) is incorrect
Choice (d) is incorrect
Consider the family of curves sketched below The gradient of these curves at the point $\left(t,y\right)$ is $8cos2t$. What is the general equation of these curves ?
 a) $f\left(t\right)=4sin2t+C$ b) $f\left(t\right)=-4sin2t+C$ c) $f\left(t\right)=-16sin2t+C$ d) $f\left(t\right)=16sin2t+C$

Choice (a) is correct!
$f\left(t\right)=\int 8cos2t\phantom{\rule{1em}{0ex}}dt=8\left(\frac{1}{2}sin2t\right)+C=4sin2t+C$
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
A family of curves have the slope $8{t}^{3}+9{t}^{2}-4t+7$ at the point $\left(t,y\right)$. Which curve in the family passes through the point (-1,2) ?
 a) $f\left(t\right)=24{t}^{2}+18t-4$ b) $f\left(t\right)=2{t}^{4}+3{t}^{3}-2{t}^{2}+7t+12$ c) $f\left(t\right)=2{t}^{4}+3{t}^{3}-2{t}^{2}+7t+2$ d) $f\left(t\right)=24{t}^{2}+18t+2$

Choice (a) is incorrect
Choice (b) is correct!
$f\left(t\right)=\int \left(8{t}^{3}+9{t}^{2}-4t+7\right)\phantom{\rule{1em}{0ex}}dt=2{t}^{4}+3{t}^{3}-2{t}^{2}+7t+C$
$f\left(-1\right)=2-3-2-7+C=2⇒C=12$,
$\text{therefore,}f\left(t\right)=2{t}^{4}+3{t}^{3}-2{t}^{2}+7t+12$.
Choice (c) is incorrect
Choice (d) is incorrect
Find the area under the curve $y=4-{x}^{2}$ between $x=-2$ and $x=2$.
 a) $0$ b) $-8$ units squared c) $\frac{32}{3}$ units squared d) $\frac{8}{3}$ units squared

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
${\int }_{-2}^{2}\left(4-{x}^{2}\right)\phantom{\rule{1em}{0ex}}dx={\left[4x-\frac{1}{3}{x}^{3}\right]}_{-2}^{2}$
$=\left(8-\frac{8}{3}\right)-\left(-8-\frac{8}{3}\right)=16-\frac{16}{3}=\frac{32}{3}.$
Choice (d) is incorrect
Find the area bounded by the curve $y={x}^{3}-{x}^{2}-2x$ and the $x$-axis.
 a) $-\frac{9}{4}$ units squared b) $\frac{37}{12}$ units squared c) $\frac{9}{4}$ units squared d) $\frac{40}{36}$ units squared

Choice (a) is incorrect
Choice (b) is correct! $\begin{array}{rcll}Area& =& A1+A2& \text{}\\ & =& {\int }_{-1}^{0}\left({x}^{3}-{x}^{2}-2x\right)\phantom{\rule{1em}{0ex}}dx-{\int }_{0}^{2}\left({x}^{3}-{x}^{2}-2x\right)\phantom{\rule{1em}{0ex}}dx& \text{}\\ & =& {\left[\frac{{x}^{4}}{4}-\frac{{x}^{3}}{3}-{x}^{2}\right]}_{-1}^{0}+{\left[\frac{{x}^{4}}{4}-\frac{{x}^{3}}{3}-{x}^{2}\right]}_{0}^{2}& \text{}\\ & =& \frac{5}{12}-\left(-\frac{8}{3}\right)& \text{}\\ & =& \frac{37}{12}.& \text{}\end{array}$
Choice (c) is incorrect
Choice (d) is incorrect
Find the area between the curves $f\left(x\right)=cosx$ and $g\left(x\right)=sinx$ between $x=0$ and $x=\frac{\pi }{4}$ (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!
${\int }_{0}^{\frac{\pi }{4}}\left(cosx-sinx\right)\phantom{\rule{1em}{0ex}}dx={\left[sinx+cosx\right]}_{0}^{\frac{\pi }{4}}$
$\phantom{\rule{56.9055pt}{0ex}}=\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}-\left(0+1\right)=\sqrt{2}-1=0.4142136\dots$
Find the area between the curves $f\left(x\right)=x+1$ and $g\left(x\right)={x}^{2}-x-2$.
 a) $\frac{32}{3}$ units squared b) $\frac{77}{3}$ units squared c) $-\frac{32}{3}$ units squared d) 24 units squared

Choice (a) is correct!
The curves intersect at $\left(-1,0\right)$ and $\left(3,4\right)$ so the area is
${\int }_{-1}^{3}\left(f\left(x\right)-g\left(x\right)\right)\phantom{\rule{1em}{0ex}}dx={\int }_{-1}^{3}\left(-{x}^{2}+2x+3\right)\phantom{\rule{1em}{0ex}}dx=\left[-\frac{{x}^{3}}{3}+{x}^{2}+3\right]=\frac{32}{3}.$
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 $2×1{0}^{4}{e}^{0.25t}$. 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) $5.044×1{0}^{7}$ b) $8.077×1{0}^{7}$ c) $1.440×1{0}^{7}$ d) $8.055×1{0}^{7}$

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
$P\left(t\right)=1{0}^{4}\left(8{e}^{0.25t}-7\right)$
$\begin{array}{rcll}\text{therefore,}Average\phantom{\rule{1em}{0ex}}value& =& \frac{1}{35-10}{\int }_{10}^{35}1{0}^{4}\left(8{e}^{0.25t}-7\right)\phantom{\rule{1em}{0ex}}dt& \text{}\\ & =& 400{\left[32\left({e}^{0.25t}-7t\right)\right]}_{10}^{35}& \text{}\\ & =& 400\left[32\left({e}^{8.75}-{e}^{2.5}\right)-7×25\right]& \text{}\\ & =& 8.055×1{0}^{7}.& \text{}\end{array}$
Which of the following statements are true?
 a) ${\int }_{-a}^{a}sinx\phantom{\rule{1em}{0ex}}dx=0$ since $sinx$ is an odd function. b) ${\int }_{-a}^{a}sinx\phantom{\rule{1em}{0ex}}dx=2{\int }_{0}^{a}sinx\phantom{\rule{1em}{0ex}}dx$ since $sinx$ is an even function. c) ${\int }_{-a}^{a}{x}^{2}\phantom{\rule{1em}{0ex}}dx=0$ since ${x}^{2}$ is an odd function. d) ${\int }_{-a}^{a}{x}^{2}\phantom{\rule{1em}{0ex}}dx=2{\int }_{0}^{a}{x}^{2}\phantom{\rule{1em}{0ex}}dx$ since ${x}^{2}$ 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