Week 7: Closure

Week 7: Closure#

Continue working on the the preparatory exercises and the in-class exercises that you have not yet completed.

Key Concepts#

  • The Riemann integral: the definite integral

  • Subdivisioning and mid-sums

  • The fundamental theorem of calculus

  • Antiderivatives: the indefinite integral

  • Partial integration and integration by substitution

  • Riemann integration of functions of two variables

  • Change of coordinates in 2D

  • Polar coordinates

If there are still concepts you are unsure about, you should reread the relevant chapters in the textbook or revisit the exercises of the week.

Extra Exercises#

We do not expect you to complete more exercises than those from from the week’s program. The following additional exercises are purely an optional offer for those who want extra practice and challenge.

1: Eight Antiderivatives That You Should Master by Now#

Provide an antiderivative of each of the following functions:

  1. \(x^n\), where \(n\) is an arbitrary constant in \(\mathbb Z\).

  2. \(x^k\), where \(k\) is an arbitrary constant in \(\mathbb Q\).

  3. \(\frac{1}{a x+b}\), where \(a\neq 0\) and \(b\) are arbitrary constants in \(\mathbb{R}\), and \(x\) is in a fitting interval.

  4. \(\cos(a x+b)\), where \(a\neq 0\) and \(b\) are arbitrary constants in \(\mathbb{R}\).

  5. \(f'(x)\), where \(f\) is differentiable.

  6. \(\sin(a x+b)\), where \(a\neq 0\) and \(b\) are arbitrary constants in \(\mathbb{R}\).

  7. \(\exp(a x+b)\), where \(a\neq 0\) and \(b\) are arbitrary constants in \(\mathbb{R}\).

  8. \(\exp(a x+b)\), where \(a\neq 0\) and \(b\) are arbitrary constants in \(\mathbb{C}\).

Answer

Bullet 1:

  • Assume that \(n>0:\) Find an antiderivative in the same way as you did in bullet 1 in Exercise I: Antiderivatives for Memorization.

  • Assume that \(n=0:\) You now need to find an antiderivative of \(1\).

  • Assume that \(n=-1:\) Since \(x^{-1}=\frac 1x,\) then we can find an antiderivative in the same was as we did in bullet 2 in Exercise I: Antiderivatives for Memorization.

  • Assume that \(n<-1:\) As an example, consider \(\int u^{-3} \mathrm{d}u=\frac{1}{-3+1}u^{-3+1}=-\frac{1}{2}u^{-2}\). This shows that \(-\frac{1}{2u^2}\) is an antiderivative to \(\frac{1}{u^3}\).

Bullet 2: Assume that \(k\neq -1:\) An antiderivative is \(\int x^{\frac pq}\mathrm{d}x=\frac{1}{\frac pq+1}x^{\frac pq+1}\).

Bullet 3: \(\frac 1a \ln(a x+b)\).

Bullet 4: \(\frac 1a \sin(a x+b)\).

Bullet 5: \(f(x)\).

Find antiderivatives to bullets 6, 7 and 8 in similar ways.

2: Parametrization of a Triangle#

Consider a set \(V \subset \mathbb{R}^2\) given by

\[\begin{equation*} V = \bigl\{ (x,y) \in \mathbb{R}^2 \big| x \ge 1 \wedge y \ge 0 \wedge x + y \le 3 \bigr\}. \end{equation*}\]

Provide a parametrization \(\pmb{r}: [0,1]^2 \to V\) of \(V\). The vector function \(\pmb{r}\) must be a function of two variables, i.e., \((u,v) \in [0,1]^2\), and the image set of \(\pmb{r}\) must be \(V\).

Note

It would be optimal if your \(\pmb{r}\) is injective on the open set \(\,]0,1[\,^2\,\), since our parametrizations must fulfill this in the upcoming weeks. But it is not a requirement here.

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