Integration

Below you will find the exercises we suggest you work on in connection with the Integration theme and the project problems which are to be done for the exercise classes during October 7-11 and October 14-18.

Section 5.1: Sums and Sigma Notation
5.1.1, 5.1.3, 5.1.7, 5.1.9, 5.1.11, 5.1.16, 5.1.17, 5.1.21, 5.1.24, 5.1.32.

Section 5.2: Areas as Limits of Sums
5.2.3, 5.2.12, 5.2.19.

Følgende Maple-regneark kan være nyttig når du skal regne ovennevnte oppgaver::Riemannsums

Section 5.3: The Definite Integral
5.3.1, 5.3.2, 5.3.9 (you may want to use the result from exercise 5.1.39 in exercise 5.3.9).

Section 5.4: Properties of the Definite Integral
5.4.1, 5.4.7, 5.4.9, 5.4.23, 5.4.27, 5.4.33, 5.4.42.

Pencast med løsning til oppgave 5.4.42

Section 5.5: The Fundamental Theorem of Calculus
5.5.1, 5.5.6, 5.5.7, 5.5.11, 5.5.14, 5.5.23, 5.5.33, 5.5.39,41, 5.5.44, 5.5.49.

Pencast med løsning til oppgave 5.5.14

Pencast med løsning til oppgave 5.5.44

Section 5.6: The Method of Substitution
5.6.1, 5.6.2, 5.6.5, 5.6.7, 5.6.8, 5.6.12, 5.6.21, 5.6.22, 5.6.45.

Pencast med løsning til oppgave 5.6.12

Pencast med løsning til oppgave 5.6.22

Følgende Maple-regneark kan være nyttig når du skal regne ovennevnte oppgaver: :Substitutions

Section 5.7: Areas of Plane Regions
5.7.1, 5.7.6, 5.7.29.

Pencast med løsning til oppgave 5.7.6 (OBS! Regnefeil i nest siste linje. Svaret skal bli 9 kvadratenheter).

These problems are to be presented during October 7-11. See here for where you should meet, and here to sign up for presenting a problem.

PDF-version of the exercises

Problem 1

a) Calculate the \(n\)'th order Taylor polynomial for \(\mathrm{e}^x\) at \(x=0\).

b) Let \(s_n =\sum_{i=0}^{n}\frac{1}{i!}.\) Use the polynomial found in a) and Lagrange's remainder to show that \[\mathrm{e}= \lim_{n \to \infty}s_n=\sum_{i=0}^{\infty}\frac{1}{i!}.\]

Problem 2

We want to show that the volume of a cone with height \(h\) and radius \(r\) is \(\pi r^2h/3 \), by approximating the volume with \(n\) cylindrical boxes with height \(h/n\), shaded green in the figure below.

a) Show that the volume of the green figure depicted in Figure 1 is \[V_{out}=\pi\left(\frac{r}{n}\right)^2\left(\frac{h}{n}\right)+\pi\left(\frac{2r}{n}\right)^2\left(\frac{h}{n}\right)+\cdots +\pi r^2\left(\frac{h}{n}\right).\]

b) Show that the volume of the green figure depicted in Figure 2 is \[V_{in}= 0+\pi\left(\frac{r}{n}\right)^2\left(\frac{h}{n}\right)+\pi\left(\frac{2r}{n}\right)^2\left(\frac{h}{n}\right)+\cdots +\pi \left(\frac{(n-1)r}{n}\right)^2\left(\frac{h}{n}\right).\]

c) Use a) and b) to prove that \(V=\pi r^2h/3\). You may find it useful that \(\sum\limits_{i=0}^{n}i^2 =n(n+1)(2n+1)/6\).

Problem 3 (Exam 1996 in 75001, problem 6)

Let \[f(x)=\frac{1}{\sqrt{1-x}}\] for \(x<1\).

a) Use induction (see page 110 in the textbook) to show that \[f^{(n)}(x)=\frac{(2n)!}{2^{2n}n!(1-x)^{(2n+1)/2}}\] for \(n=1,2,3,\dots\) (recall that \(n!=n(n-1)(n-2)\cdots 1\) and \((2n)!=2n(2n-1)(2n-2)\cdots 1\), see page 128 in the textbook).

b) Let \(P_n\) be the \(n\)th-order Taylor polynomial for \(f\) about \(a=0\) and let \(R_n(x)=f(x)-P_n(x)\). Find \(P_3(x)\) and \(R_3(x)\) and show that \[|f(x)-P_3(x)|=|R_3(x)|<5\cdot 10^{-5}\] when \(|x|<0.1\).

c) How big does \(n\) have to be in order for \[|f(x)-P_n(x)|=|R_n(x)|<5\cdot 10^{-7}\] when \(|x|<0.1\)?

Problem 4 (Exam 2000 in SIF5003, problem 9)

The equation \[x^2+y\mathrm{e}^y =1\] and the inequality \(y>-1\) uniquely defines a function \(y=f(x)\). Find \(f'(1)\), and calculate the second order Taylor polynomial for \(f\) around \(x=1\).

These problems are to be presented during October 14-18. See here for where you should meet, and here to sign up for presenting a problem.

PDF-version of the exercises

Problem 1

a) Use the substitution \(u=\sqrt[3]{x}\) to show that \[\int\frac{\sqrt[3]{x}}{\sqrt[3]{x}+1}\ dx=\int\frac{3u^3}{u+1}\ du\Big\rvert_{u=\sqrt[3]{x}}.\]

b) Use polynomial division to show that \[\frac{u^3}{u+1}=u^2-u+1-\frac{1}{u+1}.\]

c) Calculate the integral \[\int\frac{\sqrt[3]{x}}{\sqrt[3]{x}+1}\ dx.\]

Problem 2 (2: Exam 2011 in TMA4100, problem 6)

A boat lies at distance \(a\) from the wharf, and is moored with a rope at the point \(O\). A girl takes the rope, and walks along the wharf while dragging the boat by the rope, which we assume is taunt all the time. The prow of the boat follows the dotted line as shown in the figure.

We want to describe the dotted line as a graph \(y=f(x)\), for \(0< x \leq a\).

a) Show that \(y=f(x)\) satisfies \[y'=-\frac{\sqrt{a^2-x^2}}{x}.\]

b) Solve this equation by integrating it, and find the function \(f\). Use the substitution \(u=\sqrt{a^2-x^2}.\) You may also use that \[\int \frac{1}{a^2-u^2} \ du=\frac{1}{a}\ln\left(\frac{a+u}{\sqrt{a^2-u^2}}\right)+C \] for \(|u|<a.\)

Problem 3

a) Express the area of the unit disk, i.e. the area enclosed by the unit circle \(x^2+y^2 = 1\), as a definite integral.

b) Compute the definite integral in (a) using the substitution \(x = \cos(\theta)\). Is the value of the integral reasonable?

c) Use integration to find the area enclosed by the ellipse \(\frac{x^2}{9}+ \frac{y^2}{16} = 1\). (Hint: substitution, and the integral in (a).)

Problem 4 ( Exam 1993 in 75011, problem 4)

a) Find a function \(f\) so that \[\sum_{i=1}^{n}\frac{1}{n\left(2+\frac{i}{n}\right)\ln\left(2+\frac{i}{n}\right)}\] is a Riemannsum for \(f\) on the interval \([0,1]\).

b) Calculate the limit \[\lim_{n \to \infty}\sum_{i=1}^{n}\frac{1}{n\left(2+\frac{i}{n}\right)\ln\left(2+\frac{i}{n}\right)}\].