Polar Integration
We have already seen the method of u substitution and largely understood it as something of the inverse of the chain rule. This is not the whole story. Recall that I keep reiterating the idea that integration and differentiation required different definitions because, in some intrinsic way, integration applies to a broader class of functions. It is not merely an inverse operation of differentiation, but a field in its own right. Here, we begin to see why. U substitution is the beginning of the idea we will a change of variables in multivariable calculus. Our first inkling of this comes with polar integration.
1. A Geometric Definition of Polar Integration
Recall our initial definition of polar integration in terms of Reimann sums. We could have considered our volume elements in terms of polar forms. Notice that if we do that, instead of rectangles, we see triangles that are defined in terms of $r$ and $\theta$.
Worked Problems
1. A city in the shape of a circle with radius 10 miles is growing with its population density a function of the distance from the center of the city. At a distance of r miles from the center of the city its population density is $P(r) = \frac{5000}{1+r}$ people per square mile. The area of shaded rings in square miles can be approximated by the product $2\pi r \Delta r$, where r is the radius of the outer ring. Write an expression to represent the number of people in a shaded ring and find the population of the city.
$P(r) \cdot 2\pi r \Delta r =$
$\frac{5000}{1 + r} \cdot 2\pi r \Delta r =$
$\int_{r_{interior}}^{r_{exterior}} \frac{5000}{1 + r} \cdot 2\pi r dr =$
$\int_{r_{interior}}^{r_{exterior}} \frac{10000\pi \cdot r}{1 + r} dr =$
$10000\pi \left[ r – ln(r+1) \right]_{r_{interior}}^{r_{exterior}}$
$10000\pi \left[ r – ln(r+1) \right]_0^{10} =$
$10000\pi (10 – ln(10+1)) \approx 238,827 people$