Active Calculus - Multivariable

For this particular integral, regardless of the order of integration, we are unable to find an antiderivative of the integrand; in addition, even if we were able to find an antiderivative, the inner limits of integration involve relatively complicated functions.

It is therefore useful to be able to translate to other coordinate systems where the limits of integration and the evaluation of the iterated integrals are simpler. In this section we provide a brief reminder of polar coordinates, then investigate their usage for double integrals. For a more thorough discussion of polar coordinates, look at Section 9.8.

The rectangular coordinate system is best suited for graphs and regions that are naturally considered over a rectangular grid. The polar coordinate system is an alternative that offers good options for functions and domains that have more circular characteristics. A point \(P\) in rectangular coordinates that is described by an ordered pair \((x,y)\text{,}\) where \(x\) is the displacement from \(P\) to the \(y\)-axis and \(y\) is the displacement from \(P\) to the \(x\)-axis can also be described with polar coordinates \((r,\theta)\text{,}\) where \(r\) is the distance from \(P\) to the origin and \(\theta\) is the angle formed by the line segment \(\overline{OP}\) and the positive \(x\)-axis, as shown at left in Figure 12.5.2.

Trigonometry and the Pythagorean Theorem allow for straightforward conversion from rectangular to polar, and vice versa.

Remember that you will need to interpret the results of \(r\) as being positive or negative along with interpreting the quadrant of \(\theta\text{.}\) You can look at Example 9.8.12 and later examples for different ways to interpret and make conversions between rectangular and polar coordinates.

We can draw graphs of curves in polar coordinates in a similar way to how we do in rectangular coordinates. However, when plotting in polar coordinates, we use a grid that considers changes in angles and changes in distance from the origin. In particular, the angles \(\theta\) and distances \(r\) partition the plane into small wedges as shown in Figure 12.5.2.

To make the change to polar coordinates, we not only need to represent the variables \(x\) and \(y\) in polar coordinates, but we also must understand how to write the area element, \(dA\text{,}\) in polar coordinates. That is, we must determine how the area element \(dA\) can be written in terms of \(dr\) and \(d\theta\) in the context of polar coordinates.

Remember that our classic calculus approach to defining the double integral involved breaking each coordinate into smaller pieces which created a collection of smaller areas that we used to create our Riemann sum. We can do the same procedure in terms of polar coordinates (instead of rectangular coordinates) and we could have our smaller pieces be in terms of some small changes in \(r\) and \(\theta\) (like the regions on the polar grid in Figure 12.5.2). We will now need to understand how to measure the area of a piece described by \(R_0\leq r \leq R_1 \quad \theta_0 \leq \theta \leq \theta_1\text{,}\) as shown in Figure 12.5.4.

We only have the second step of our conversion left above: convert the region of integration described by inequalities that correspond to fixing one coordinate and bounding the other in a consistent way. Similar to how we used vertically simple and horizontally simple descriptions for rectangular regions, we want to set up inequalities by hold one of our coordinates constant and setting boundary functions for the other coordinate.

Each of the types of regions above corresponds to exactly the necessary description to use a iterated integral. Let’s look at the inequalities and plot for a radially simple region; For any fixed value of \(\theta\) in the interval \([a,b]\text{,}\) we want to look at a slice along this constant \(\theta\) value. This corresponds to a ray from the origin in the cone between the grey lines in Figure 12.5.6. Along this \(\theta=k\) ray, the region of interest has an upper and lower bound on the \(r\)-coordinate. In fact, there is the function that describes the upper boundary, \(r=g_2(\theta)\text{,}\) for every value of \(\theta\) in \([a,b]\text{.}\) Similarly, there is the function that describes the lower boundary, \(r=g_1(\theta)\text{,}\) for every value of \(\theta\) in \([a,b]\text{.}\) Geometrically, this means that along every gray, dashed slice (where \(\theta\) is constant), there is the same upper and lower boundary curves, shown in red and blue, respectively.

While angularly simple regions are used much less often than radially simple regions, a parallel structure is used; Along every slice with a constant value of \(r\) in \([c,d]\) (the gray, dashed curves in Figure 12.5.7), the upper bound on \(\theta\) is given by the function \(\theta=h_2(r)\) and the lower bound is given by \(\theta=h_1(r)\) (shown by the red and blue bounding curves in Figure 12.5.7).

Just as in the rectangular coordinate case, especially complicated regions may need to be split into several different regions that are radially or angularly simple.

Note again that the order of integration corresponds to the manner in which we have sliced the region of integration, with the variable being held constant along a slice being the last variable to be integrated.

While there is no firm rule for when polar coordinates can or should be used, they are a natural alternative anytime the domain of integration may be expressed simply in polar form, and/or when the integrand involves expressions such as \(\sqrt{x^2 + y^2}\text{.}\)

In the next activity, we will convert an equation from rectangular to polar coordinates and use this polar equation to give bounds of integration for two different integrals.