Exercise 1: Seven Antiderivatives You Must Know by Heart
A
For which of the following functions of $x$, that are given on appropriate domains, can you immediately state an antiderivative (an indefinite integral)?
If you get stuck, then use the “think backwards” technique. You already know a lot of differentiation rules
by heart. When performing elementary integration, ask yourself if you can think of a function that when differentiated becomes the function that you are trying to integrate. That must be an antiderivative.
answer
If you had to pass on any of the above, then find an antiderivative using Maple’s int.
Please store the results in your long-term memory. These must be known by heart.
Exercise 2: Seven Integration Rules You Must Master
A
State an antiderivative of each of the following functions of $x$ defined on appropriate domains:
It is (not always but often) advantageous to do this rewriting from fraction to power form. Then you can integrate fractions as well without having to memorize more integration rules.
When you solve the rest of today’s exercises, refer back to exercise 1 and 2 when needed for the integration rules. When the day is over, you will (must!) have these antiderivatives stored firmly in your memory, ready for retrieval at any moment.
Exercise 3: Computational Rules for Indefinite Integrals
In this and the following exercise we will work on examples of some important building blocks within integral calculus: Sequences of numbers and their possible convergence.
From The Great Danish Encyclopedia, translated (Gyldendal):
“convergence, concept of fundamental importance in mathematical analysis, in particular in the theory for infinite series. A sequence of real numbers $x_1,x_2,\ldots$ is called convergent, if a number $x$ exists, such that the number $x_n$ is arbitrarily close to $x$, when $n$ is sufficiently large (…). The number $x$ is called the limit value of the sequence, and the sequence is said to converge towards $x\,.$ If the sequence is not convergent, it is divergent.”
A
Four sequences $\,\lbrace a_n\rbrace\,,$$\,\lbrace b_n\rbrace\,,$$\,\lbrace c_n\rbrace\,$ and $\,\lbrace d_n\rbrace\,$ are for $n\in \mathbb N$ given by
Decide which of these four sequences that are convergent, and state the limit values for those that are convergent.
hint
Consider for each what happens to the value of the number when $n\to \infty$.
hint
The limit value is difficult to figure out when you have $n$ in both the numerator and denominator. For $\lbrace b_n\rbrace$ you might have to split the fraction into two terms.
hint
For $\lbrace d_n\rbrace$, after splitting the fraction in two, one of the two new fractions might still be troublesome. Try multiplying through in both numerator and denominator with a fitting term.
answer
$\lbrace a_n \rbrace\,$ is convergent with limit value $0$.
$\lbrace b_n \rbrace\,$ is convergent with limit value $\frac 12\,.$
$\lbrace c_n \rbrace\,$ is divergent.
$\lbrace a_n \rbrace\,$ is convergent with limit value $-\frac 43\,.$
Exercise 5: Integrals Using Left Sums
In this exercise you will determine integration values using so-called left sums. The technique involves subdividing the area under the graph of the function into columns, and then summing up their areas as their number goes towards infinity.
For that, we state (without proof) the following formula for the sum of an arithmetic series, which might come in handy:
corresponding to a subdivision of the interval $[0\,,\,1]$ in $\,n\,$ segments of equal size.
hint
A left sum is the sum of a number of rectangle (column) areas below the graph of $f\,$ whose left corners touch the graph.
hint
A rectangle area is “length times width”. Write out an expression where you sum this up for all columns. Then reduce using the fact that all column widths are equal.
hint
Since the line is straight, the difference in area between two consecutive rectangles is constant.
State the centre and the radius of $C$. Choose a parametric representation $\,\mathbf r(u)\,$ of $C$ corresponding to one passage around the circle. Determine the Jacobian that corresponds to your parametric representation.
hint
When creating your parametrization, don’t forget to define the interval of the parameter.
answer
Centre is at $(0,1,1)\,,$ and the radius is $2$.
A possible parametrization (among many correct options) is
$$\mathbf r(u)=(2\cos(u),2\sin(u)+1,1)\quad , \quad u \in [0,2\pi].$$
Its Jacobian is then
$$\text{Jac}(u)=|\mathbf r'(u)|=2.$$
B
Plot the parametrization in Maple.
In general, when creating a parametrization, always plot it while you are creating it to visually verify your result. Often, when a parametrization is just a bit off, the visual impact is dramatic. Also, plotting it will help you to not forget to also define the parameter intervals.
C
Given the function $\,f(x,y,z)=x^2+y^2+z^2\,,$ determine the restriction $\,f(\mathbf r(u))\,.$ Then compute the line integral
Does the line integral depend on the parametric representation you chose for the circle? Test other parametric representations and compute the line integral based on these. You could for instance change direction of the “passage”/”drawing direction” by changing parameter signs within the trigonometric functions, or you could change the “passage speed” by adjusting both coefficients and interval. (After any such adjustment, check the plot to ensure that you are still drawing the same geometry and did not make a mistake.)
answer
No, a line integral does not depend on the chosen parametric representation.
E
Does the line integral depend on the location of the circle? Try to displace the circle by $1$ upwards (parallel to the $y$ axis), and compute the line integral again.
answer
Yes, a line integral might depend on the location of the object.
F
Discuss the above two results with a fellow student. Why is this the case?
Exercise 9: Arc Length Using Mid-Point Sums (Advanced)
eNote 23 uses left sums throughout. But it is also possible to use right sums or even mid-point sums! We will try a mid-point-sum approach in this exercise to compute the length of the parabolic arc segment
Assume that the interval $\left[\,0,1\,\right]$ is subdivided into $n$ segments of equal length $\delta u$, and let the division points be denoted $u_i$ as in eNote 23. Connect the points on the parabola vertically above the division points with straight line segments (parabola chords). Then the sum of the lengths of the chords will be a good approximation of the arc length.
Show that the sum of chord lengths can be expressed as
