MS-A0211 - Differential and Integral Calculus 2, Lecture, 10.1.2023-20.2.2023
This course space end date is set to 20.02.2023 Search Courses: MS-A0211
Lecture 10 - 9.2.2023
Completion requirements
In this lecture we studied:
- Double integrals. Polar coordinates. We recalled the definition of polar coordinates. As an example, we computed the integral from -infity to -infinity of exp(-x^2) by relating it of the double integral over the whole xy-plane of exp(-x^2 -y^2). This later calculation can be done in polar coordinates but is impossible to do directly in cartesian coordinates.
- Area of a region. We defined the area of a 2-dimensional region, that is, the doble integral over the region of f(x,y)=1. As an example, we computed the area of a disk of radius a.
- General changes of variables. The transformation to polar coordinates is a particular case of change of variables. We discussed changes of variable in general and that the change in area of a ''infinitesimal'' area element dA is given by multiplication by the absolute value of the determinant of the Jacobian matrix of the transformation. As an example, we computed the area of a elliptic disk E. For that, we used a change of variable so that E turns into a disk.
- Center of mass. We introduced the concept of center of mass. In the one variable case, starting with the case of discrete masses and then for a continuous mass distribution by using Riemann sums, we derived the integral formula for the center of mass. We derived the center of mass formula in 2D. For that, we have to compute the total mass and the total moments. We did an example.
(image from Wikipedia) The notions studied in this lecture can be found in section 14.4 (Polar coordinates and change of variables) of Adam and Essex’s Calculus book (seventh edition) and section 15.3 (Moment and Center of Mass) of D. Guichard and friends's book.