Question

If possible, write iterated integrals in cylindrical coordinates for the following regions in the specified orders. Sketch the region of integration. The region outside the cylinder \(r=1\) and inside the sphere \(\rho=5,\) for \(z \geq 0,\) in the orders \(d z d r d \theta, d r d z d \theta,\) and \(d \theta d z d r\)

Short answer

#Answer 1. The boundaries of the region are $1 \leq r \leq \sqrt{\rho^2 - z^2}$, $z \geq 0$ and $0 \leq \theta \leq 2\pi$. Finally, the sphere has a radius of 5, so we have $0 \leq z \leq \sqrt{25-r^2}$. 2. The iterated integral in the order $d z d r d \theta$ is $\int_{0}^{2\pi} \int_{1}^{\sqrt{25-z^2}} \int_{0}^{\sqrt{25-r^2}} f(r,z,\theta) \,dz\, dr\, d\theta$. 3. The iterated integral in the order $d r d z d \theta$ is $\int_{0}^{2\pi} \int_{0}^{\sqrt{25-r^2}} \int_{1}^{\sqrt{25-z^2}} f(r,z,\theta) \,dr\, dz\, d\theta$. 4. The iterated integral in the order $d \theta d z d r$ is $\int_{1}^{\sqrt{25-z^2}} \int_{0}^{\sqrt{25-r^2}} \int_{0}^{2\pi} f(r,z,\theta) \,d\theta\, dz\, dr$.

Step-by-step solution

1. Determine the boundaries of the region

First, let's convert the equation of the sphere from spherical to cylindrical coordinates: $$ r^2+z^2 = \rho^2 \implies z = \sqrt{\rho^2 - r^2} $$ The cylinder has a radius of 1, so the region is bounded by \(r = 1\). Therefore, the boundaries of the region are \(1 \leq r \leq \sqrt{\rho^2 - z^2}\), \(z \geq 0\) and \(0 \leq \theta \leq 2\pi\). Finally, the sphere has a radius of 5, so we have \(0 \leq z \leq \sqrt{25-r^2}\).

2. Write the iterated integral in the order \(d z d r d \theta\)

In this order, we first integrate with respect to \(z\), then \(r\), and finally \(\theta\). The integral will be: $$ \int_{0}^{2\pi} \int_{1}^{\sqrt{25-z^2}} \int_{0}^{\sqrt{25-r^2}} f(r,z,\theta) \,dz\, dr\, d\theta $$

3. Write the iterated integral in the order \(d r d z d \theta\)

In this order, we first integrate with respect to \(r\), then \(z\), and finally \(\theta\). The integral will be: $$ \int_{0}^{2\pi} \int_{0}^{\sqrt{25-r^2}} \int_{1}^{\sqrt{25-z^2}} f(r,z,\theta) \,dr\, dz\, d\theta $$

4. Write the iterated integral in the order \(d \theta d z d r\)

In this order, we first integrate with respect to \(\theta\), then \(z\), and finally \(r\). The integral will be: $$ \int_{1}^{\sqrt{25-z^2}} \int_{0}^{\sqrt{25-r^2}} \int_{0}^{2\pi} f(r,z,\theta) \,d\theta\, dz\, dr $$

Key concepts

Cylindrical Coordinates

Cylindrical coordinates are a popular method for analyzing problems that have symmetry around a central axis. They are often preferred in scenarios where circular symmetry is present, like in the example involving a cylinder and a sphere. Cylindrical coordinates consist of three parameters:

• \( r \): This represents the radial distance from the origin to a point in the plane.
• \( \theta \): This denotes the angular position measured from a reference direction, often the positive x-axis.
• \( z \): This is the height above the plane.

Using cylindrical coordinates can greatly simplify the computation of integrals over circular regions or volumes since they neatly align with the geometry of these shapes.
This is why they were used in the example to convert the region inside a sphere but outside a cylinder into a simpler form. Cylindrical coordinates allow us to easily express the limits of integration by clearly distinguishing between the radial, angular, and vertical extents of a volume.

Multivariable Calculus

Multivariable Calculus extends the concepts of calculus to functions of multiple variables. It is essential for analyzing and understanding phenomena involving more than one dimension. In the given problem, we deal with three variables: radius \( r \), angle \( \theta \), and height \( z \). Multivariable calculus techniques allow us to analyze how these variables interact to define a volume in space. Calculating integrals of such volumes requires advanced understanding, as we must take into account changes in each variable and how they affect the overall result of the integral.
By using iterated integrals, a form of multiple integrals, we are able to compute volumes and other multi-dimensional physical quantities systematically.
This is crucial when dealing with real-world 3-dimensional objects, allowing precise calculations through mathematical expressions.

Triple Integrals

Triple integrals extend the concept of integration to three dimensions, allowing for the calculation of volumes, masses, and other quantities over a 3-dimensional region. In the context of the example, triple integrals take the form of nested integrations. We process each dimension's contribution separately, venturing through each subsequent integral. The general form of a triple integral in cylindrical coordinates looks like: \[\int \int \int f(r, \theta, z) \, dz \, dr \, d\theta\]The integral evaluates the function \( f(r, \theta, z) \) over the bounds specified for each variable.
Triple integrals are powerful tools when analyzing complex volumes with distinct mathematical behaviors in multiple dimensions. They allow for precise determination of a region's properties by handling the limits of integration carefully for each variable independently.

Integration Bounds

In multivariable calculus, accurately defining the integration bounds is crucial. These bounds determine the region over which the function will be integrated and are essential for correctly setting up an integral. The given solution demonstrates calculating three different iterated integrals, each with distinct integration orders:

• In each case, \( \theta \) varies from 0 to \( 2\pi \), rendering a full circle.
• \( r \) is bounded between 1 and the projection of the sphere onto the plane, \( \sqrt{25-z^2} \).
• \( z \) stretches from 0 to \( \sqrt{25-r^2} \), covering the height from the bottom plane up to the sphere.

By setting these bounds appropriately, the integral correctly encompasses the region defined between the cylinder and the sphere. Understanding how to establish these boundaries ensures that the integration process accurately reflects the spatial constraints imposed by the geometry of the problem.