Question

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

Short answer

The region is a spherical sector with radius 2, cone angle \( \pi/4 \), and 360-degree azimuthal coverage.

Step-by-step solution

Understanding Spherical Coordinates

In spherical coordinates, any point in space is defined by three parameters: \( \rho \) (the distance from the origin), \( \theta \) (the azimuthal angle in the \( xy \)-plane), and \( \varphi \) (the polar angle from the positive \( z \)-axis). The triple integral given is expressed as \( \int \int \int \rho^{2} \sin(\varphi) \, d\rho \, d\theta \, d\varphi \), which represents a volume integral in spherical coordinates.

Analyzing the Limits for \( \theta \)

The outer integral runs from \( 0 \) to \( 2\pi \), which denotes the full rotation around the \( z \)-axis, or equivalently, a complete circle in the \( xy \)-plane. This covers the entire 360-degree surround.

Analyzing the Limits for \( \varphi \)

The middle integral runs from \( 0 \) to \( \pi/4 \). The angle \( \varphi \) starts from the positive \( z \)-axis and extends to an angle of \( \pi/4 \) radians. This confines our region to a cone with its apex at the origin and opening along the \( z \)-axis, extending to \( \pi/4 \) radians from that axis.

Analyzing the Limits for \( \rho \)

The inner integral is from \( 0 \) to \( 2 \), which specifies that \( \rho \), the radial distance from the origin, ranges from the origin to a length of 2 units. Therefore, any point within this region is at most 2 units away from the origin.

Describing the 3D Region

Combining the above, the limits define a spherical sector of a sphere (a cone-like segment), with a cone angle of \( \pi/4 \) radians in the vertical direction, full 360-degree coverage around the \( z \)-axis, and extending radially up to 2 units away from the origin. This region resembles a small 'cap' or 'section' of a spherical shell.

Key concepts

Spherical Coordinates

Spherical coordinates offer a different method from Cartesian coordinates to define a point in space. This system utilizes three parameters:

• \( \rho \): The radial distance from the origin, indicating how far the point is from the center.
• \( \theta \): The azimuthal angle, which describes the direction in the \( xy \)-plane. It ranges from \( 0 \) to \( 2\pi \), allowing for full circular coverage.
• \( \varphi \): The polar angle, which starts from the positive \( z \)-axis and measures downwards. This angle helps define its vertical position.

Spherical coordinates are particularly helpful when dealing with problems that exhibit radial symmetry about a point, such as those involving spheres and cones. Because they naturally resonate with circular or spherical geometries, they simplify integrations over round shapes in three-dimensional spaces.
This integration method is immensely powerful for calculating volumes, as it gracefully handles transformations from flat \( xy \)-planes into curvilinear 3D forms.

Volume Integral

The concept of a volume integral is to integrate a function over a three-dimensional region. In spherical coordinates, a volume integral often involves expressions like \( \rho^2 \sin(\varphi) \), which account for the spherical geometry.
When computing a volume integral in spherical coordinates, the integrand usually starts with \( \rho^2 \sin(\varphi) \), which captures the Jacobian determinant. This factor ensures that distance and area distortions are correctly addressed as you move outwards from the origin.In our specific integral, \(\int_{0}^{2 \pi} \int_{0}^{\pi / 4} \int_{0}^{2} \rho^2 \sin(\varphi) \, d\rho \, d\theta \, d\varphi\), we are articulating a space defined by a full rotation around the \( z \)-axis and extending from the \( z \)-axis outward. Through this integral, we essentially create a 'slice' of a sphere, defining it from the radial center of the sphere to its outer boundary at 2 units and within a cone of \( \pi/4 \) radians.

Polar Angle

The polar angle \( \varphi \) is one of the essential elements in spherical coordinates and holds a unique significance. Starting from the positive \( z \)-axis, \( \varphi \) measures how far a point drops downward.
Within our current integral bounds of \( 0 \) to \( \pi/4 \), this angle captures a specific conical section. Specifically, as \( \varphi \) increases from 0 to \( \pi/4 \), it describes a cone whose tip is located at the origin and opens upwards along the \( z \)-axis.This conical section mirrors several natural phenomena we encounter in physics and engineering, such as the shapes of sound cones or coverage regions of radar. Understanding how these angles modify and shape three-dimensional regions significantly impacts our ability to calculate volumes and shapes efficiently in dynamic and complex systems.