Week 7: Preparation

Week 7: Preparation#

Reading Material#

  • Reading: Chapter 6 through Section 6.3

  • Python: Demo 7

Key Concepts#

Long Day will cover:

  • The Riemann integral: the definite integral

  • Limits

  • Subdivisioning and mid-sums

  • The fundamental theorem of calculus

  • Antiderivatives: the indefinite integral

Short Day will cover:

  • Integration by parts

  • Integration by substitution

  • Riemann integration of functions of two variables

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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