Question

A triple integral in spherical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{2 \pi} \int_{\pi / 6}^{\pi / 4} \int_{0}^{2} \rho^{2} \sin (\varphi) d \rho d \theta d \varphi $$

Short answer

The region is a spherical cap between angles 30° and 45° from the z-axis, with radius 2.

Step-by-step solution

Identify the Integral Components

The given triple integral is \(\int_{0}^{2 \pi} \int_{\pi / 6}^{\pi / 4} \int_{0}^{2} \rho^{2} \sin (\varphi) d \rho d \theta d \varphi \). The integral is in spherical coordinates where \( \rho \) represents the radial distance, \( \theta \) is the azimuthal angle, and \( \varphi \) is the polar angle.

Describing the \(\rho\) Boundary

The innermost integral is with respect to \(\rho\), which represents the distance from the origin. It ranges from 0 to 2, meaning we are considering all points within a sphere of radius 2, centered at the origin.

Describing the \(\varphi\) Boundary

The middle integral is with respect to \(\varphi\), the polar angle, ranging from \(\pi/6\) to \(\pi/4\). This limits the region to a spherical cap, covering slices from around 30 degrees to 45 degrees down from the positive z-axis.

Describing the \(\theta\) Boundary

The outermost integral is with respect to \(\theta\), the azimuthal angle, ranging from 0 to \(2\pi\). This accounts for a full rotation around the z-axis, meaning the region defined is symmetric about the z-axis.

Combine Bounds to Define the Region

The combined bounds describe a segment of a sphere: a spherical cap ranging from angles 30 degrees to 45 degrees down from the z-axis, with a radius up to 2, symmetric around the z-axis covering a full circle.

Key concepts

Spherical Coordinates

Spherical coordinates are an extension of the polar coordinate system into three dimensions. They are particularly useful for describing regions in space that have symmetry around a point, such as spheres or spherical caps. In this system, any point in space is defined by three values:

• \(\rho\): the radial distance from the origin.
• \(\theta\): the azimuthal angle, a horizontal angle measured in the plane perpendicular to the z-axis.
• \(\varphi\): the polar angle, the angle measured from the positive z-axis downward.

This system is highly effective when applying calculus to 3D problems where surfaces or volumes have circular symmetry. Remember that converting between Cartesian and spherical coordinates involves the following relationships:

• \(x = \rho \sin(\varphi) \cos(\theta)\)
• \(y = \rho \sin(\varphi) \sin(\theta)\)
• \(z = \rho \cos(\varphi)\)

Spherical Cap

A spherical cap is a portion of a sphere "cut" by a plane. When studying integrals in spherical coordinates, defining the boundaries of such shapes becomes crucial. In the context of the exercise, the region of integration forms a spherical cap, limited by specific angles and a radius.This cap is defined by:

• A polar angle \(\varphi\) between \(\pi/6\) and \(\pi/4\), which describes the vertical slice of the sphere from 30 to 45 degrees off the z-axis.
• An azimuthal angle \(\theta\) that spans from 0 to \(2\pi\), allowing for a full rotation around the z-axis, capturing all possible azimuthal directions.
• A radial distance \(\rho\) extending from 0 to 2, signifying the size of the sphere being considered.

Understanding the concept of a spherical cap helps visualize the spatial region of integration as a "cap" that sits atop a spherical volume, defined by angular and radial constraints.

Polar Angle

The polar angle, denoted as \(\varphi\), is crucial for defining vertical positioning within spherical coordinates. It measures the angle from the positive z-axis down to the point of interest.

• In our integral, \(\varphi\) ranges from \(\pi/6\) to \(\pi/4\), meaning that it calculates slices from 30 degrees to 45 degrees below the z-axis. This slice is part of what forms the spherical cap we are interested in.
• The smaller the angle, the closer to the top of the sphere, which is aligned with the z-axis. As \(\varphi\) increases, the slice moves further down towards the xy-plane.

Being familiar with the polar angle's role in spherical coordinates is essential for determining how a region spans vertically across a spherical space. This understanding is key to solving integrals involving spherical symmetry.

Azimuthal Angle

The azimuthal angle, represented by \(\theta\), plays a significant role in defining horizontal positioning within spherical coordinates. It's measured in the plane parallel to the xy-plane, determining the rotation around the z-axis.

• In practice, it ranges from 0 to \(2\pi\) within our integral, symbolizing a complete rotation. This ensures that the described region maintains symmetry about the z-axis.
• The angle starts at the positive x-axis and measures around the z-axis counterclockwise.

Understanding \(\theta\) is essential for effectively translating between spherical and Cartesian coordinates, especially when involving integration. By controlling the range of the azimuthal angle in an integral, we define the horizontal span of the region of interest, crucial for problems involving full rotational symmetry.