0

This question comes up when I read the Page 4 of MIT Mathematics OpenCourseWare 18.01 lecture note.

In a polar coordinates system, when calculating the area of the function $r = f(\theta)$, why did the lecture note choose a "right triangle" to approximate the small slice's area? Two edges around the right angle are: $r$ and $d\theta$, and the slice area is $\frac{1}{2} r^2 d\theta$.

Why not use a sector of circle (even though it gives the same result)? Why not use a isosceles triangle?

To extend the above specific question to a more abstract question: How do we know that the approximation for a small slice can give us an accurate result after integrating? and Is there another course that shows the strict proof if this course does not reach that rigidity level?

Dachuan Huang
  • 151
  • 1
    slides you have shared are very long. better reformulate your problem. Give specific example here and ask specific question – Lee May 19 '21 at 05:54
  • I think that to be completely rigorous the approximation should include a bound on the error, and it should be shown that the error converges to zero as $d\theta$ converges to zero. That said, I think that often calculus 1 courses are more focused on knowing how to "do" calculus, than on rigorously proving every result, which is more typical in an analysis course. Although, a good calculus textbook probably shows a fairly rigorous derivation. – Joe May 19 '21 at 07:39
  • Take those notes as a table of formulas, but ignore the explanations. It is more accurate to say that the area is given by that formula by definition. The justification starts from defining area as the integral of $1$, in Cartesian coordinates, then showing that changing to polar coordinate makes a factor $r$ appear that is the Jacobian determinant of the transformation. Finally applying Fubini's theorem to integrate with respect to $r$. – plop May 19 '21 at 10:48
  • The difference between a right triangle and a scalene triangle is of second order, product of $\frac12 \cdot dr \cdot r d\theta $ which can be neglected. – Narasimham May 19 '21 at 16:14

0 Answers0