Week 7: Preparation

Week 7: Preparation#

Key Concepts#

After reading, you should be able to explain the following 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

This week, we will explore these key concepts in great detail. We expect you to have familiarized yourself with these topics before lectures.

Reading Material#

We recommend that you read the textbook. Watching YouTube videos on the week’s topics can be useful, but it should not replace proper preparation for the week’s program and is not recommended as a standalone approach.

Read and study the following:

  • Reading: Sections 6.1 through 6.3

  • Python demo07

Preparatory Exercises#

I: Antiderivatives for Memorization#

For which of the following functions can you immediately provide an antiderivative?

  1. \(x^n, n \in \mathbb{N}\)

  2. \(\frac{1}{x}\)

  3. \(\ln(x)\)

  4. \(\frac{1}{1+x^2}\)

  5. \(\cos(x)\)

  6. \(\sin(x)\)

  7. \(\exp(x)\)

Where you had to give up, find an antiderivative using SymPy’s integrate and please store the result in your long-term memory.

II: Riemann Sum for \(x^2\)#

Consider a function \(f: [0,1] \to \mathbb{R}\) given by

\[\begin{equation*} f(x)=x^2. \end{equation*}\]

Compute the Riemann sum of \(f\) over \([0,1]\) with 4 same-sized subintervals using the right end-points in each subinterval. Then, compare the result with the exact value of the integral, which can be found using the antiderivative.

Answer

The Riemann sum gives

\[\begin{equation*} \sum_{j=1}^4 \left(\frac{j}{4}\right)^2 \frac{1}{4} = 15/32 \approx 0.47. \end{equation*}\]

The exact value found using the antiderivative is \(1/3\).

III: Parametrization of a Disc Boundary#

Let

\[\begin{equation*} B = \{ (x,y) \in \mathbb{R}^2 \mid x^2 + y^2 \le 2\} \end{equation*}\]

be a circular disc in the plane centred at \((0,0)\) with a radius of \(\sqrt{2}\). Provide a parameterization of its boundary \(\partial B\).

Answer

There are infinitely many ways to parameterize the boundary. Here is a reasonable choice using polar coordinates:

\[\begin{equation*} \pmb{r}(t) = (\sqrt{2} \cos(t), \sqrt{2}\sin(t)) \quad \text{for } t \in [0, 2 \pi[. \end{equation*}\]

Here is a less reasonable parameterization:

\(\pmb{r}_1(x) = (x, \sqrt{2-x^2})\) for \(x \in [-\sqrt{2}, \sqrt{2}]\) which must be combined with \(\pmb{r}_2(x) = (x, -\sqrt{2-x^2})\) for \(x \in ]-\sqrt{2}, \sqrt{2}[\).

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