1.1 Definition of the Integral

Draw a rectangle that encompasses the entire shaded area, and one that is encompassed by the shaded area. The shaded area is no more than the area of the bigger rectangle, and no less than the area of the smaller rectangle.

The area is between $1.5$ and $2.5$ square units.

We can improve on the method of Question 1 by using three rectangles that together encompass the shaded region, and three rectangles that together are encompassed by the shaded region.

The shaded area is between 2.75 and 4.25 square units. (Other estimates are possible, but this is a reasonable estimate, using methods from this chapter.)

Four rectangles suffice.

The area under the curve is a number in the interval $\left( \frac{3}{8}\left[\frac{1}{2}+\frac{1}{\sqrt{2}}\right], \frac{3}{8}\left[1+\frac{1}{\sqrt{2}}\right]\right)$.

Try drawing a picture.

left

Try an oscillating function.

Many answers are possible. One example is $f(x)=\sin x$, $[a,b]=[0,\pi]$, $n=1$. Another example is $f(x)=\sin x$, $[a,b]=[0,5\pi]$, $n=5$.

The ordering of the parts is intentional: each sum can be written by changing some small part of the sum before it.

Some of the possible answers are given, but more exist.

If we raise $-1$ to an even power, we get $+1$, and if we raise it to an odd power, we get $-1$.

Some answers are below, but others are possible.

Sometimes a little anti-simplification can make the pattern more clear.

(a), (b) These are geometric sums.
(c) You can write this as three separate sums.
(d) You can write this as two separate sums. Remember that $e$ is a constant. Don't be thrown off by the index being $n$ instead of $i$.

Since the sum adds four pieces, there will be four rectangles. However, one might be extremely small.

Write out the general formula for the left Riemann sum from Definition 1.1.11 in the CLP-2 text and choose $a$, $b$ and $n$ to make it match the given sum.

$n=4$, $a=2$, and $b=6$

Since the sum runs from 1 to 3, there are three intervals. Suppose $2 = \Delta x = \frac{b-a}{n}$. You may assume the sum given is a right Riemann sum (as opposed to left or midpoint).

One answer is below, but other interpretations exist.

Let $\Delta x = \dfrac{\pi}{20}$. Then what is $b-a$?

Many interpretations are possible–see the solution to Question 13 for a more thorough discussion–but the most obvious is given below.

Notice that the index starts at $k=0$, instead of $k=1$. Write out the given sum explicitly without using summation notation, and sketch where the rectangles would fall on a graph of $y=f(x)$.

Then try to identify $b-a$, and $n$, followed by “right”, “left”, or “midpoint”, and finally $a$.

Three answers are possible. It is a midpoint Riemann sum for $f$ on the interval $[1,5]$ with $n =4$. It is also a left Riemann sum for $f$ on the interval $[1.5,5.5]$ with $n =4$. It is also a right Riemann sum for $f$ on the interval $[0.5,4.5]$ with $n =4$.

The area is a triangle.

$\dfrac{25}{2}$

There is one triangle of positive area, and one of negative area.

$\dfrac{21}{2}$

Review Definition 1.1.11 in the

CLP-2 text.

$\sum\limits_{i=1}^{50} \Big(5+\big(i-\nicefrac{1}{2}\big)\frac{1}{5}\Big)^8 \ \frac{1}{5}$

$54$

You'll want the limit as $n$ goes to infinity of a sum with $n$ terms. If you're having a hard time coming up with the sum in terms of $n$, try writing a sum with a finite number of terms of your choosing. Then, think about how that sum would change if it had $n$ terms.

$\displaystyle\int_{-1}^{7}f(x)\,\,\dee{x}=\displaystyle\lim_{n\to\infty}\displaystyle\sum_{i=1}^{n} f\left(-1+\frac{8i}{n}\right)\frac{8}{n}$

The main step is to express the given sum as the right Riemann sum,

Don't be afraid to guess $\De x$ and $f(x$) (review Definition 1.1.11 in the

CLP-2 text). Then write out explicitly $\sum\limits_{i=1}^{n} f(a+i\De x)\Delta x$ with your guess substituted in, and compare the result with the given sum. Adjust your guess if they don't match.

$f(x) = \sin^2 (2 + x)$ and $b=4$

The main step is to express the given sum as the right Riemann sum $\sum\limits_{k=1}^{n} f(a+k\De x)\Delta x$. Don't be afraid to guess $\De x$ and $f(x$) (review Definition 1.1.11 in the

CLP-2 text). Then write out explicitly $\sum\limits_{k=1}^{n} f(a+k\De x)\Delta x$ with your guess substituted in, and compare the result with the given sum. Adjust your guess if they don't match.

$f(x)=x\sqrt{1-x^2}$

The main step is to express the given sum in the form $\sum_{i=1}^{n} f(x_i^*)\Delta x$. Don't be afraid to guess $\De x$, $x_i^*$ (for either a left or a right or a midpoint sum — review Definition 1.1.11 in the

CLP-2 text) and $f(x$). Then write out explicitly $\sum_{i=1}^{n} f(x_i^*)\Delta x$ with your guess substituted in, and compare the result with the given sum. Adjust your guess if they don't match.

$\int_0^3 e^{-x/3}\cos(x)\,\,\dee{x}$

The main step is to express the given sum in the form $\sum\limits_{i=1}^{n} f(x_i^*)\Delta x$. Don't be afraid to guess $\De x$, $x_i^*$ (probably, based on the symbol $R_n$, assuming we have a right Riemann sum — review Definition 1.1.11 in the CLP-2 text) and $f(x$). Then write out explicitly $\sum\limits_{i=1}^{n} f(x_i^*)\Delta x$ with your guess substituted in, and compare the result with the given sum. Adjust your guess if they don't match.

$\displaystyle\int_0^1 x e^{x}\,\,\dee{x}$

Try several different choices of $\De x$ and $x_i^*$.

Possible answers include:
$\displaystyle\int\limits_0^2 e^{-1-x}\ \,\dee{x}$, $\displaystyle\int\limits_1^3 e^{-x}\ \,\dee{x}$, $2\displaystyle\int_{1/2}^{3/2} e^{-2x}\ \,\dee{x}$, and $2\displaystyle\int\limits_0^1 e^{-1-2x}\ \,\dee{x}$.

Let $x=r^3$, and re–write the sum in terms of $x$.

$\dfrac{r^{3n+3}-1}{r^3-1}$

Note the sum does not start at $r^0=1$.

$r^5\left(\dfrac{r^{96}-1}{r-1}\right)$

Draw a picture. See Example 1.1.15 in the

CLP-2 text.

$5$

Draw a picture. Remember $|x| = \left\{\begin{array}{rc}x&x\ge 0\\-x&x<0\end{array}\right.$ .

16

Draw a picture: the area we want is a trapezoid. If you don't remember a formula for the area of a trapezoid, think of it as the difference of two triangles.

$\dfrac{b^2-a^2}{2}$

You can draw a very similar picture to Question 30, but remember the areas are negative.

$\dfrac{b^2-a^2}{2}$

If $y=\sqrt{16-x^2}$, then $y$ is nonnegative, and $y^2+x^2=16$.

$4\pi$

Sketch the graph of $f(x)$.

$\displaystyle\int_0^3 f(x)\,\,\dee{x} = 2.5$

At which time in the interval, for example, $0\le t\le 0.5$, is the car moving the fastest?

53 m

What are the possible speeds the car could have reached at time $t=0.25$?

true

You need to know the speed of the plane at the midpoints of your intervals, so (for example) noon to 1pm is not one of your intervals.

3200 km

Sure looks like a Riemann sum.

(a) There are many possible answers. Two are $\int_{-2}^0 \sqrt{4-x^2}\,\,\dee{x}$ and $\int_0^2 \sqrt{4-(-2+x)^2}\,\,\dee{x}$. (b) $\pi$

For part (b): don't panic! Just take it one step at a time. The first step is to write down the Riemann sum. The second step is to evaluate the sum, using the given identity. The third step is to evaluate the limit $n\rightarrow\infty$.

(a) $30$ (b) $41 \frac{1}{4}$

The first step is to write down the Riemann sum. The second step is to evaluate the sum, using the given formulas. The third step is to evaluate the limit as $n\rightarrow\infty$.

$\dfrac{56}{3}$

The first step is to write down the Riemann sum. The second step is to evaluate the sum, using the given formulas. The third step is to evaluate the limit $n\rightarrow\infty$.

$6$

You've probably seen this hint before. It is worth repeating. Don't panic! Just take it one step at a time. The first step is to write down the Riemann sum. The second step is to evaluate the sum, using the given formula. The third step is to evaluate the limit $n\rightarrow\infty$.

$12$

Using the definition of a right Riemann sum, we can come up with an expression for $f(-5+10i)$. In order to find $f(x)$, set $x=-5+10i$.

$f(x)=\dfrac{3}{10}\left(\dfrac{x}{5}+8\right)^2\sin\left(\dfrac{2x}{5}+2\right)$

Recall that for a positive constant $a$, $\diff{}{x}\left\{a^x\right\} = a^x \log a$, where $\log a$ is the natural logarithm (base $e$) of $a$.

$\dfrac{1}{\log 2}$

Part (a) follows the same pattern as Question 43–there's just a little more algebra involved, since our lower limit of integration is not 0.

(a) $\dfrac{1}{\log 10}\left(10^b-10^a\right)$
(b) $\dfrac{1}{\log c}\left(c^b-c^a\right)$; yes, it agrees.

Your area can be divided into a section of a circle and a triangle. Then you can use geometry to find the area of each piece.

$\frac{\pi}{4} -\frac{1}{2} \arccos(a) + \frac{1}{2}a\sqrt{1-a^2}$

Since $f(x)$ is linear, there exist real numbers $m$ and $c$ such that $f(x)=mx+c$. It's a little easier to first look at a single triangle from each sum, rather than the sums in their entirety.

true (but note, for a non-linear function, it is possible that the midpoint Riemann sum is not the average of the other two)