# Quiz 1: Lower and Upper Sums

## Question 1

The graph of a function $f$
is shown below. Area ${A}_{1}=3.6$,
area ${A}_{2}=1.1$ and
area ${A}_{3}=0.7$.
What is ${\int}_{0.5}^{6}\phantom{\rule{0.3em}{0ex}}f\left(x\right)\phantom{\rule{0.3em}{0ex}}dx$ ?

**Your answer is correct**

Areas are always positive quantities, unlike definite integrals which can
be sums of positive and negative terms. In this case, the integral equals
${A}_{1}-{A}_{2}+{A}_{3}$.

**Not correct. You may try again.**

Remember that areas are always positive, but definite integrals can be sums of
positive and negative terms. Think about how to combine the three areas to arrive at
the correct value of the definite integral.

## Question 2

Find U, the Riemann Upper Sum for $f\left(x\right)={x}^{2}$
on $\left[0,2\right]$, using 4 equal
sub-intervals.

**Your answer is correct.**

**Not correct. Choice (b) is false.**

Remember that to find U, you must use the maximum value of
$f\left(x\right)$ on each
sub-interval.

**Not correct. Choice (c) is false.**

Remember that to find U, you must use the maximum value of
$f\left(x\right)$ on
each sub-interval.

**Not correct. Choice (d) is false.**

Remember that to find U, you must use the maximum value of
$f\left(x\right)$ on
each sub-interval.

**Not correct. Choice (e) is false.**

## Question 3

Find L, the Riemann Lower Sum for $f\left(x\right)={x}^{2}$
on $\left[0,2\right]$, using 4 equal
sub-intervals.

**Not correct. Choice (a) is false.**

Remember that to find L, you must use the minimum value of
$f\left(x\right)$ on
each sub-interval.

**Not correct. Choice (b) is false.**

Remember that to find L, you must use the minimum value of
$f\left(x\right)$ on
each sub-interval.

**Your answer is correct.**

**Not correct. Choice (d) is false.**

Remember that to find L, you must use the minimum value of
$f\left(x\right)$ on
each sub-interval.

**Not correct. Choice (e) is false.**

## Question 4

When calculating the Riemann Upper and Lower Sums (U and L) for the function
$f\left(x\right)={x}^{2}$ on the
interval $\left[0,2\right]$,
what is the smallest number of (equal) sub-intervals needed to make
$U-L\le 0.1$ ?

**Not correct. Choice (a) is false.**

This function is an increasing function on
$\left[0,2\right]$, and so the difference
between U and L when $n$
equal sub-intervals are used is just the length of each sub-interval times the difference
between $f\left(2\right)$
and $f\left(0\right)$.

**Not correct. Choice (b) is false.**

This function is an increasing function on
$\left[0,2\right]$, and so the difference
between U and L when $n$
equal sub-intervals are used is just the length of each sub-interval times the difference
between $f\left(2\right)$
and $f\left(0\right)$.

**Not correct. Choice (c) is false.**

This function is an increasing function on
$\left[0,2\right]$, and so the difference
between U and L when $n$
equal sub-intervals are used is just the length of each sub-interval times the difference
between $f\left(2\right)$
and $f\left(0\right)$.

**Your answer is correct.**

**Not correct. Choice (e) is false.**

## Question 5

The function $f$
is a continuous function defined on the interval
$\left[a,b\right]$, where
$a<b$. Read
all the statements below and tick all those that are correct.

**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.

Remember that area is always a positive quantity, by definition.

**There is at least one mistake.**

For example, choice (c) should be true.

**There is at least one mistake.**

For example, choice (d) should be false.

When
$f\left(t\right)$ is positive,
${\int}_{a}^{c}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt\phantom{\rule{1em}{0ex}}\le \phantom{\rule{1em}{0ex}}{\int}_{a}^{b}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt$ regardless of
the values of $f\left(c\right)$
and $f\left(b\right)$.

**There is at least one mistake.**

For example, choice (e) should be false.

**Your answers are correct**

**True**.**False**. Remember that area is always a positive quantity, by definition.**True**.**False**. When $f\left(t\right)$ is positive, ${\int}_{a}^{c}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt\phantom{\rule{1em}{0ex}}\le \phantom{\rule{1em}{0ex}}{\int}_{a}^{b}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt$ regardless of the values of $f\left(c\right)$ and $f\left(b\right)$.**False**.

## Question 6

Estimate the value of ${\int}_{1}^{3}lnx\phantom{\rule{0.3em}{0ex}}\phantom{\rule{1em}{0ex}}dx$ using
the Riemann Upper Sum U for $f\left(x\right)=lnx$
on the interval $\left[1,3\right]$,
with 4 equal sub-intervals.

**Not correct. Choice (a) is false.**

Have you used
the correct function values in your expression for U? For example, on the first sub-interval
$\left[1,\frac{3}{2}\right]$, the maximum
function value is $ln\frac{3}{2}.$

**Not correct. Choice (b) is false.**

Have you used
the correct function values in your expression for U? For example, on the second sub-interval
$\left[\frac{3}{2},2\right]$, the maximum
function value is $ln2.$

**Your answer is correct.**

**Not correct. Choice (d) is false.**

Have you used
the correct function values in your expression for U? For example, on the third sub-interval
$\left[2,\frac{5}{2}\right]$, the maximum
function value is $ln\frac{5}{2}.$

**Not correct. Choice (e) is false.**

## Question 7

Given that $cos\sqrt{x}$ decreases
on the interval $\left[0,9\right]$,
estimate the value of ${\int}_{0}^{9}cos\sqrt{x}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}dx$
using the Riemann Lower Sum L on this interval with three unequal sub-intervals
$\left[0,1\right],\phantom{\rule{0.3em}{0ex}}\left[1,4\right],\phantom{\rule{0.3em}{0ex}}\left[4,9\right].$ Enter
your answer correct to two decimal places.

**Your answer is correct**

Well done!

**Not correct. You may try again.**

Is your calculator set to radian mode? If not, change it.

## Question 8

Water leaks out of a tank at a decreasing rate. The rate of outflow is monitored every
five minutes for twenty minutes, and the results are tabulated below.
$$\begin{array}{cccccc}\hfill \text{Time(min)}\hfill & \hfill 0\hfill & \hfill 5\hfill & \hfill 10\hfill & \hfill 15\hfill & \hfill 20\hfill \\ \hfill \text{Outflow(litres/min)}\hfill & \hfill 10\hfill & \hfill 9\hfill & \hfill 8\hfill & \hfill 6\hfill & \hfill 4\hfill \\ \hfill \hfill \end{array}$$
One estimate for the number of litres of water lost over the 20 minute period is
the average of L and U for the function describing this loss. Which option
equals the average of L and U?

**Not correct. Choice (a) is false.**

Remember that for a decreasing
function, the minimum function value on a sub-interval occurs at the right
hand end and the maximum value occurs at the left-hand end.

**Not correct. Choice (b) is false.**

Remember that for a decreasing function, the minimum function value on a
sub-interval occurs at the right hand end and the maximum value occurs
at the left-hand end.

**Not correct. Choice (c) is false.**

Remember that for a decreasing function,
the minimum function value on a sub-interval occurs at the right hand
end and the maximum value occurs at the left-hand end.

**Your answer is correct.**

Here
$U=5\left(10+9+8+6\right)=165$ and
$L=5\left(9+8+6+4\right)=135$ and so the
average is $150$
litres/minute.

## Question 9

The function $g$
whose graph appears below has these values at the indicated values of
$t$:
$$\begin{array}{ccccccccc}\hfill t\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 2\hfill & \hfill 3\hfill & \hfill 4\hfill & \hfill 5\hfill & \hfill 6\hfill & \hfill 7\hfill \\ \hfill g\left(t\right)\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill -2\hfill & \hfill -3\hfill & \hfill -2\hfill & \hfill 0\hfill \\ \hfill \hfill \end{array}$$
What is the value of the lower Riemann sum for
$g\left(t\right)$ on
$\left[0,7\right]$? (Use
7 equal intervals of length 1.)

**Your answer is correct**

Well done!

**Not correct. You may try again.**

Obtain the minimum function values on each sub-interval from the values
given in the table and by inspecting the graph. For example, on the sub-interval
$\left[4,5\right]$, the minimum
value is $f\left(5\right)$.

## Question 10

Suppose that $U$ and
$L$ are the upper and lower
Riemann sums for $f\left(t\right)$
on the interval $\left[a,b\right]$,
using $n$
equal subdivisions. Read the two statements and then select the correct
option.

(1) For all functions $f$, $U$ and $L$ are never equal. In fact, $L$ is always less than $U$.

(2) For all functions $f$, ${\int}_{a}^{b}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt\phantom{\rule{1em}{0ex}}<\phantom{\rule{1em}{0ex}}U$.

(1) For all functions $f$, $U$ and $L$ are never equal. In fact, $L$ is always less than $U$.

(2) For all functions $f$, ${\int}_{a}^{b}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt\phantom{\rule{1em}{0ex}}<\phantom{\rule{1em}{0ex}}U$.

**Not correct. Choice (a) is false.**

For statement (1), for example, think about the way
$L$ and
$U$ are
defined, in terms of the minimum and maximum values (respectively) of
the function on each sub-interval. For which type of function could they be
equal?

**Not correct. Choice (b) is false.**

For statement (1), for example, think about the way
$L$ and
$U$ are
defined, in terms of the minimum and maximum values (respectively) of
the function on each sub-interval. For which type of function could they be
equal?

**Not correct. Choice (c) is false.**

For (2), think about what would happen if
$f\left(t\right)=1$ for all
$t$.

**Your answer is correct.**

The statements are not true for all functions. For example, when
$f\left(t\right)$ is constant on
the interval $\left[a,b\right]$,
$U=L={\int}_{a}^{b}\phantom{\rule{0.3em}{0ex}}f\left(t\right)\phantom{\rule{0.3em}{0ex}}dt.$
However, for non-constant functions, both statements are true.