Deriving polar substitution formula for double integrals

Question

I'm studying double integrals (the theory part, not the problem solving). We're trying to derive the formula for the polar substitution in a double integral:

$$\iint_{D}f(x, y)dxdy = \iint_{D_1}f(r\cos\theta, r\sin\theta)rdrd\theta$$ where $x = r\cos\theta, y = r\sin\theta$ maps $D_1 \to D$

First I want to say that it is explicitly stated in the textbook that we're not going to prove the general formula for substitutions in the integral (the one that uses the Jacobian) but rather, just for now, the formula for the polar substitution.

It starts by defining a set $S := \big\{(r, \theta)|\bar{r_1} \leq r\leq \bar{r_2}, \bar{\theta_1} \leq \theta \leq \bar{\theta_2}\big\}$ and dividing this set using lines/curves $r = r_j$ and $\theta = \theta_j$ where: $$r_j := \bar{r_1} + j\triangle r, \text{where } \triangle r := \frac{\bar{r_2} - \bar{r_1}}{n} \\\theta_j:= \bar{\theta_1}+j\triangle \theta, \text{where } \triangle\theta := \frac{\bar{\theta_2} - \bar{\theta_1}}{n} $$

and $j=0, 1, ..., n$

This part is clear, it's pretty similar to dividing a rectangular region when defining a double integral itself.

So an illustration would look something like this (sorry for the bad picture, couldn't find a way to do it in Desmos):

The following sentence really confuses me because it's stated as a given (without any explanation whatsoever):

If we look at the region $\big\{ (r, \theta)| r_0 \leq r \leq r_0 + \triangle r, \theta_0 \leq \theta \leq \theta_0 + \triangle \theta \big\}$ we notice that the area of that region is $$\frac{1}{2}\triangle \theta ((r_0 + \triangle r)^2 - r_0^2) = (r_0 + \frac{\triangle r}{2})\triangle r \triangle \theta$$

The region that we're talking about is, as far as I can see, the 'bottom-right' sub-region of the set $S$, since $r_0 = \bar{r_1}, \theta_0 = \bar{\theta_1}$. What I don't see is how what they said is the area of that region? Speaking in polar terms the region defined should actually be a rectangle, so I guess the area should be just $\triangle r \triangle \theta$, but apparently I'm missing something. After this statement it goes on to show the formula I've mention above and it's all clear from here on, it's just this one part that I can't understand.

Any ideas?

Thanks.

Answer 1

The area of that polar "rectangle" comes from subtracting two concentric sectors of circles with radii $r_0+\Delta r$ and $r_0$ respectively, both subtending the same arc $\Delta \theta$ at their common centre.

Note that when measuring angles in radians, the area of such a sector is given by $$\frac12 r^2\phi,$$ where $r$ is the radius of the sector, and $\phi$ its central angle. This comes from assuming that the area of a sector of a circle is proportional to the arc of that sector. A circle may be considered an arc of $2π$ radians. Thus, if the area wanted is $A,$ we have that $$\frac{A}{πr^2}=\frac{\phi}{2π}.$$ This gives the formula stated above.