Suppose we have a integrable region \(S\) in \(\mathbb{R}^3\). To find it's volume, we need to compute \(\int_S 1\) via iterated integration. In order to do so, we want to choose and easily computable order of integration. Below, two possible integration orderings are implemented for several different \(S\).
Moving a slider will translate a plane through \(S\) and show the intersection. For example, consider the ordering \(\iiint \text{d}c \text{d}b \text{d}a \). Moving the \(a\)-slider translates a plane perpendicular to the \(a\)-axis and shows the blue intersection between it and \(S\). Subsequently moving the \(b\)-slider translates a plane perpendicular to the \(b\)-axis and shows the intersection between it, the \(a\)-axis plane, and \(S\).
Below is the solution to \(1 \leq z \leq x^2 + y^2\), a solid cone. Play with both integration orders to get a feeling for the kinds of slices they produce. Determine which integration ordering is the best.
Hint: If you are having trouble deciding which is a nicer ordering, try writing the integral on paper using each ordering.
| Axes: | |
| XZ-Plane: | |
| Volume Opacity = 0 | |
| Intersection Plane: |
Here, we have the solution to \( \frac{1}{2}x^2 + y^2 + z^2 \leq 1 \), an ellipsoid. Play with the sliders to find the best ordering. In the process, abstract a useful heuristic for recognizing good slices by considering both this example and the previous.
Here, we have the solution to
$$ \begin{cases} 1 &\geq x^2 + y^2 \\ 1 &\geq y^2 + z^2 \end{cases} $$which is, in fact, the intersection of two slid cylinders (conventionally called the Steinmetz solid). As with the previous examples, find the best order of integration for this region. Make use of the pattern's you've noticed so far, and relate them to the symmetries of the volume itself.