\\\\(
\nonumber
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Week 4: Long Day: The Riemann Integral and Its Use on Curves
Today we will dive into on of the main subjects in the course syllabus: Integration with multiple variables, a topic that will unfold over the rest of the Spring semester.
We start off from the symbol
$$\int_a^b f(x)\,\mathrm{d}x$$
that you know already from highschool. We will today clarify the meaning of the symbol more precisely as a limit value of a sum. Normally we have two ways of computing this limit value: either by directly computing the limit value or by using an indefinite integral on $f$. These approaches can be generalized to two and three variables. We will work out how to find the integral along a curve described by a curvilinear coordinate system. Here we will look at parametrizations and the Jacobian. It turns out that the tangent vector plays an important part in this context.
Today’s Key Concepts
Limit value $\lim$. The Riemann integral $\int_a^b$. Double sum and double integral. Triple sum and triple integral. Parametrization of curve. Tangent vector and tangent to parametrized curve. The Jacobian $\mathrm{Jac}$.
Preparation and Syllabus
Today we dive into eNote 23 Riemann Integrals about the Riemann integral on functions of one and two variables. We will also browse through eNote 24 Line and Plane Integrals when we need insight into parametrized curves - see in particular the examples of parametrization of various geometric shapes.
Maple Ressources
int
computes integrals, both definite and indefinite.
Today’s Maple demo is 26_LineIntegrals.
Activity Program
- 10.00 – 12.00: $\,$ Lecture (aud. 42, b. 303A) (link to streaming)
- 12.00 – 17.00: $\,$ Group exercises in the study areas (b. 302, bottom floor)
- 13.00 – 16.00: $\,$ Your teachers are present
Group Exercises
- Seven Antiderivatives You Must Know by Heart
- Seven Integration Rules You Must Master
- Computational Rules for Indefinite Integrals
- Sequences of Numbers
- Integral as a Limit Value of a Left Sum
- Use of the Fundamental Theorem. By Hand
- The Tangent Vector and the Length of a Parametrized Curve
- Parametrization and the Line Integral. By Hand
- Arc Length Using the Midpoint Sum (Advanced)