I'm currently reading Calculus by Thomas and I cant seem to understand the argument for the following double integral in polar form formula .
The text says that the double integral of a function over a region R in polar coordinates is defined as:
$$\iint_R f(r,θ) dA = lim_{n\to \infty} \sum_{k=1}^n f(r_k, θ_k) \Delta A_k$$
where $\Delta A_k$ is the area of the kth polar rectangle when we divide the region R into n polar rectangles.
The text then says that the area of each polar rectangle $\Delta A_k$ = $r_k$ $\Delta r$ $\Delta θ$ as per the following image from the book:
which gives
$$\iint_R f(r,θ) dA = lim_{n\to \infty} \sum_{k=1}^n f(r_k, θ_k)r_k \Delta r \Delta θ$$
The text then says as $n\to \infty$ and $\Delta r\to 0$ and $\Delta θ\to 0$, the sum converges to
$$lim_{n\to \infty} \sum_{k=1}^n f(r_k, θ_k)r_k \Delta r \Delta θ =\iint_R f(r,θ)r dr dθ $$
Lastly the text says, " A version of Fubini's theorem says that the limit approached by these sums can be evaluated by single integrations with respect to r and θ as
$$\iint_R f(r,θ) dA = \int_{θ=a}^{θ=b} \int_{r = g_1(θ)}^{r= g_2(θ)} f(r,θ)r dr dθ $$ "
My question is the integral in the last statement inferred from the riemann sum of the polar rectangles * height of the function
or is it inferred like in case of rectangular coordinates where the inner integral represents the cross sectional area or does it come about as a particular case of a general change of variables.
Best Answer
Of the three choices you present, the reasoning is most nearly that of the case of rectangular coordinates. However, there are two subtleties to watch out for, which at this point in calculus Thomas quite properly glosses over since they would just confuse the early student.
The first is that it is very possible to have a region which cannot be described by assigning a simply-connected range in $r$ to each value of $\theta$. For such a region, you can still do the integral using the same general technique, but you may have to break it up into multiple regions first.
The other trap is that if the function $f(r,\theta)$ osscilates from positive to negative, and osscilates very rapidly in some part of the region such that $$\iint_R |f(r,\theta)|\,dA $$ diverges, then doing the integral in an $r$-first (or for that matter $\theta$-first) manner might artificially select among several possible values for the integral. Again, consideration of such pathological cases is better left for a later course in Analysis.