Question

If possible, write iterated integrals in spherical coordinates for the following regions in the specified orders. Sketch the region of integration. Assume that \(f\) is continuous on the region. \(\int_{0}^{2 \pi} \int_{\pi / 6}^{\pi / 2} \int_{\csc \varphi}^{2} f(\rho, \varphi, \theta) \rho^{2} \sin \varphi d \rho d \varphi d \theta\) in the orders \(d \rho d \theta d \varphi\) and \(d \theta d \rho d \varphi\)

Short answer

Question: Rewrite the given integral in the orders \(d \rho d \theta d \varphi\) and \(d\theta d \rho d \varphi\): Given integral: $\int_{0}^{2 \pi} \int_{\pi / 6}^{\pi / 2} \int_{\csc \varphi}^{2} f(\rho, \varphi, \theta) \rho^{2} \sin \varphi d \rho d \varphi d \theta$ Answer: 1. \(d \rho d \theta d \varphi\) order: \(\int_{\pi / 6}^{\pi / 2} \int_{0}^{2\pi} \int_{\csc \varphi}^{2} f(\rho, \varphi, \theta) \rho^{2} \sin \varphi d \rho d \theta d \varphi\) 2. \(d\theta d \rho d \varphi\) order: \(\int_{\pi / 6}^{\pi / 2} \int_{\csc \varphi}^{2} \int_{0}^{2\pi} f(\rho, \varphi, \theta) \rho^{2} \sin \varphi d \theta d \rho d \varphi\)

Step-by-step solution

Analyze the given integral in spherical coordinates

Given integral: $\int_{0}^{2 \pi} \int_{\pi / 6}^{\pi / 2} \int_{\csc \varphi}^{2} f(\rho, \varphi, \theta) \rho^{2} \sin \varphi d \rho d \varphi d \theta$ In spherical coordinates, the order of integration is \(d \rho d \varphi d \theta\). We can see that: - \(\theta\) varies from \(0\) to \(2 \pi\), which is a full revolution around the \(z\)-axis. - \(\varphi\) varies from \(\pi / 6\) to \(\pi / 2\), which means that the region lies in the azimuthal angle between \(30^\circ\) and \(90^\circ\). - \(\rho\) varies from \(\csc \varphi\) to \(2\). This bound depends on the angle \(\varphi\). Now, let's rewrite the given integral with the other orders of integration.

Rewrite the integral with \(d\rho d\theta d\varphi\) order

To convert the integral to the order \(d \rho d \theta d \varphi\), we need to express the bounds of \(\theta\) in terms of \(\rho\) and \(\varphi\), but in this case, no such transformation is needed because \(\theta\) has fixed bounds from \(0\) to \(2 \pi\). The new integral is: $\int_{\pi / 6}^{\pi / 2} \int_{0}^{2\pi} \int_{\csc \varphi}^{2} f(\rho, \varphi, \theta) \rho^{2} \sin \varphi d \rho d \theta d \varphi$

Rewrite the integral with \(d\theta d\rho d\varphi\) order

Finally, we'll rewrite the integral in the order \(d \theta d \rho d \varphi\). As in step 2, no transformation is needed for the bounds of \(\theta\). The new integral is: \(\int_{\pi / 6}^{\pi / 2} \int_{\csc \varphi}^{2} \int_{0}^{2\pi} f(\rho, \varphi, \theta) \rho^{2} \sin \varphi d \theta d \rho d \varphi\) So the iterated integral in the orders \(d \rho d \theta d \varphi\) and \(d\theta d \rho d \varphi\) are: 1. \(d \rho d \theta d \varphi\) order: \(\int_{\pi / 6}^{\pi / 2} \int_{0}^{2\pi} \int_{\csc \varphi}^{2} f(\rho, \varphi, \theta) \rho^{2} \sin \varphi d \rho d \theta d \varphi\) 2. \(d\theta d \rho d \varphi\) order: \(\int_{\pi / 6}^{\pi / 2} \int_{\csc \varphi}^{2} \int_{0}^{2\pi} f(\rho, \varphi, \theta) \rho^{2} \sin \varphi d \theta d \rho d \varphi\)

Key concepts

Iterated Integrals

To truly grasp what an iterated integral is in the context of spherical coordinates, let's start by defining the term. An iterated integral is simply a sequence of integrals, typically used to evaluate a function over a multi-dimensional region. In spherical coordinates, this involves integrating a function over a sphere or a portion of a sphere. Imagine peeling an onion, layer by layer, but with math! Each integral in the sequence focuses on a particular dimension:

• The first integral evaluates over a radius (\(\rho\)),
• The second applies to polar angles \(\varphi\)
• and the third considers the azimuthal angle \(\theta\)
.

By integrating in stages, we build up a complete evaluation of the function over a specified region. Including the factor \(\rho^2 \sin \varphi\) is crucial when integrating in spherical coordinates because it accounts for the volume element at each step. Each order of integration reorders these stages, but results in the same volume or value, like rearranging the shelves in a library without changing the books.

Integration Orders

Understanding integration orders in spherical coordinates is like navigating a three-dimensional map, where the sequence of steps taken can vary. In our given problem, we're asked to switch up this sequence. The order initially provided is \(d \rho d \varphi d \theta\). This means we integrate over the radial distance \(\rho\) first, then the polar angle \(\varphi\), and finally the azimuthal angle \(\theta\).Another possible order is \(d \rho d \theta d \varphi\), where now \(\theta\) follows \(\rho\), and \(\varphi\) comes last. The bounds for \(\theta\) and \(\varphi\) remain constant, but \(\rho\) still switches based on \(\varphi\).A third order proposed, \(d \theta d \rho d \varphi\), tackles \(\theta\) first, letting it complete a full circle from 0 to \(2\pi\) before moving inside. The integration order doesn't change the end result because the function integrates over the exact same area, but it could ease computation or fit the symmetry of the function in question better.

Spherical Coordinate Bounds

Spherical coordinate bounds might sound technical, but they're just the numerical limits within which we integrate. Think of them as the start and endpoint of a journey through the spherical coordinate system. In our example, the bounds are defined as follows:

• For \(\theta\), the azimuthal angle, it varies from 0 to \(2 \pi\) which represents a complete rotation around the z-axis.
• For \(\varphi\), the polar angle, it's limited from \(\pi/6\) (or 30 degrees) to \(\pi/2\) (or 90 degrees), covering a quadrant of the sphere.
• The radial distance \(\rho\) is bounded between \(\csc \varphi\) and 2. Here, \(\csc \varphi\) adjusts depending on the angle \(\varphi\), creating a dynamic lower boundary, while 2 is a fixed upper boundary. This defines a shell-like region between two spherical surfaces.

Correctly adjusting these bounds for each integration order ensures that all potential paths through the volume are captured, no matter the route.