Point 3: $\,\displaystyle{\frac 1a \,\ln(a\cdot x+b)}\,.$
Point 4: $\,\displaystyle{\frac 1a \,\sin(a\cdot x+b)}\,.$
In the same way with points 5, 6 and 7.
Exercise 3: Computational Rules for Indefinite Integrals
Intro: In this and the following exercise we give samples on important building blocks in integral calculus : Sequences of number and their possible convergence. From The Great Danish Encyclopedia (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 $x_n$ are arbitrarily close to $x$, when $n$ is sufficiently large $(\ldots)$. The number $x$ is called he limit value for the sequence, that is said to converge towards $x\,.$ If the sequence is not convergent, it is called divergent.’’
A
Four sequences $\,\left{a_n\right}\,,$$\,\left{b_n\right}\,,$$\,\left{c_n\right}\,$ and $\,\left{d_n\right}\,$ are for $n\in \mathbb N$ given by
Decide which of these four sequences that are convergent, and state (without calculation) the limit values for those that are convergent.
answer
$\,\left{a_n\right}\,$ is convergent with the limit value 0.
$\,\left{b_n\right}\,$ is convergent with the limit value $\frac 12\,.$$\,\left{c_n\right}\,$ is divergent.
$\,\left{a_n\right}\,$ is convergent with the limit value $-\frac 43\,.$
Exercise 5: Integrals Using Left Sums
A
State a left sum $\,V_n\,$ for the Function
$$\,f(x)=x\,,\,\,x\in \left[\,0\,,\,1\,\right]$$
corresponding to a tesselation af the interval [$\,0\,,\,1\,$] in $\,n\,$ segments of equal length. Determine using this
$$\,\displaystyle{\int_0^1 x\,\mathrm{d}x}\,.$$
hint
The left sum you get as the sum of a number of areas of rectangles below the graph for $f\,.$ The difference between the area of two consequtive rectangles is constant, therefore the areas of the rectangles constitute a difference series. The sum of $n$ consequtive terms in a difference series is given by the formula:
$$ S=\frac n2(a_1+a_n)\,.$$
hint
The rectangels has the areas $\displaystyle{0\,\,,\,\,\frac{1}{n^2}\,\,,\,\,\frac{2}{n^2}\,\,,\,\,\frac{3}{n^2}\,\ldots\,\frac{n-1}{n^2}}\,.$
Determine the tangent vector for $\mathbf r(u)$ in the point $(2,1)\,$ and the length of the tangent vector. Plot the curve together with the tangent vector.
hint
Does the the point lie on the curve at all? Find $u_0$ so that $\mathbf r(u_0)=(2,1)\,,$ it is not difficult.
answer
$u_0=1\,.$
The wanted tangent vector $\mathbf r’(1)=(4,3)\,.$
The length is $|\,\mathbf r’(1)\,|=5\,.$
B
How long is the part of the curve that corresponds to $u\in \left[\,0,1\,\right]\,$ and how long is the part of the curve that corresponds to $u\in \left[\,1,2\,\right]\,?$
hint
Integrate the Jacobi-function for $\,\mathbf r\,$ between 0 and 1. Use Maple to find an indefinite integral.
answer
The first part has the length $\frac{61}{27}\approx 2.26\,.$
The second part has the length $\frac{104}{27}\,\sqrt{13}-\frac{125}{27}\approx 9.27\,.$
Exercise 8: Parametrization and Curve Integral
By Hand:
In the $(x,y,z)$-space we consider the circle C given by
State the centre and the radius for C. Chose a parametric representation $\,\mathbf r(u)\,$ for C corresponding to one passage of the circle. Determine the Jacobi-function corresponding to the parametric representation.
answer
Centre is in $(0,1,1)\,.$
E.g. $\mathbf r(u)=(2\cos(u),2\sin(u)+1,1)\,.$
Jacobi$(u)=|\mathbf r’(u)|=2\,.$
B
Given the function $\,f(x,y,z)=x^2+y^2+z^2\,.$ Determine the restriction $\,f(\mathbf r(u))\,,$ and determine the curvilinear integral
Is the curvilinear integral dependent of the parametric representation you chose for the circle? Test other parametric representations and compute the curvilinear integral based on these. E.g. you can change direction of the passage or the passage speed.
D
Does the curvilinear integral depend on the position of the circle? E.g. try to displace the circle 1 parallel to the direction of the $y$-axis, and compute the curvilinear integral once more.
Exercise 9: Arc Length Using the Mid-Point Sum
In eNote 23 we use left sums everywhere, since the functional values used are taken in the left endpoint of the interval. But it is also possible to use the midpoints of the intervals, when one states the sum, by which you obtain a so-called midpoint sum. We use this in the present problem where we must 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 to the arc length. Show that the sum of lengths of the chords can be expressed as
