Question

A triple integral in spherical coordinates is given. Describe the region in space defined by the bounds of the integral. $$ \int_{0}^{\pi / 2} \int_{0}^{\pi} \int_{0}^{1} \rho^{2} \sin (\varphi) d \rho d \theta d \varphi $$

Short answer

The region is a spherical cap of radius 1, covering the northern hemisphere portion from \( \theta = 0 \) to \( \theta = \pi \).

Step-by-step solution

Identify the Order of Integration

The integral is in the form \( \int_{0}^{\pi / 2} \int_{0}^{\pi} \int_{0}^{1} \rho^{2} \sin (\varphi) d \rho \, d \theta \, d \varphi \). This means we integrate first with respect to \( \rho \), then \( \theta \), and finally \( \varphi \).

Determine the Range of \( \rho \)

The innermost integral is \( \int_{0}^{1} \), which tells us that \( \rho \) varies from 0 to 1. In spherical coordinates, \( \rho \) represents the radial distance from the origin to any point in the region.

Determine the Range of \( \theta \)

The middle integral is \( \int_{0}^{\pi} \), indicating that \( \theta \) ranges from 0 to \( \pi \). In spherical coordinates, \( \theta \) is the azimuthal angle, measuring the angle around the \( z \)-axis.

Determine the Range of \( \varphi \)

The outermost integral is \( \int_{0}^{\pi / 2} \), showing that \( \varphi \) ranges from 0 to \( \pi / 2 \). Here, \( \varphi \) is the polar angle, measuring the angle down from the positive \( z \)-axis.

Describe the Region in Space

The described region in space is a spherical cap. It is a portion of a sphere of radius 1, centered at the origin. The cap covers the northern hemisphere (\( 0 \leq \varphi \leq \pi/2 \)) and spans from \( \theta = 0 \) to \( \theta = \pi \).

Key concepts

Triple Integral

A triple integral is a powerful mathematical tool used to calculate the volume of a three-dimensional region. In this context, we are focusing on a triple integral in spherical coordinates, which is particularly useful for integrating over regions defined by spheres or spherical surfaces.

The specified triple integral in the original exercise is \[ \int_{0}^{\pi / 2} \int_{0}^{\pi} \int_{0}^{1} \rho^{2} \sin (\varphi) d \rho \, d \theta \, d \varphi \]. Here, each integral bound signifies the limits for three different spherical coordinates:

• \( \rho \): the radial distance from the origin,
• \( \theta \): the azimuthal angle,
• \( \varphi \): the polar angle.

Each level of integration helps to piece together the entire volume, accounting for the radial, angular, and polar dimensions of the space the integral describes.

Spherical Cap

A spherical cap is a portion of a sphere cut off by a plane. In this exercise, the region defined by the triple integral is a spherical cap. We are considering a sphere centered at the origin with a radius of 1.

Because the radial distance \( \rho \) ranges from 0 to 1, the region considered extends outwards only as far as a distance of 1 unit, which forms part of a sphere's surface. The polar angle \( \varphi \) ranges from 0 to \( \pi/2 \), which indicates that the cap is in the northern hemisphere, covering from the positive pole down to the equator of the sphere.

The bounds for \( \theta \) from 0 to \( \pi \) imply that the cap sweeps halfway around the sphere horizontally, covering all points within \( z = 0 \) horizontally up to the top of the sphere. It's a three-dimensional region that can be visualized as a dome resting on the top half of the sphere.

Polar Angle

The polar angle, denoted as \( \varphi \), is one of the angles used to locate a point in spherical coordinates. The polar angle measures the inclination from the positive \( z \)-axis toward the point, essentially determining its height relative to the \( xy \)-plane.

In the given integral, \( \varphi \) ranges from 0 to \( \pi/2 \). This range indicates that the points considered start directly above the origin, on the positive \( z \)-axis, and extend down to the equatorial plane of the spherical region.

Understanding the polar angle is key to visualizing the symmetry of the space and the vertical distribution of points that form the three-dimensional volume under consideration. It helps carve out that dome-like part of the sphere known as the spherical cap.

Azimuthal Angle

The azimuthal angle, often represented by \( \theta \), is another essential component in spherical coordinate systems. It measures the angle around the \( z \)-axis, similar to the concept of longitude in geographic coordinates.

In our current exercise, the azimuthal angle \( \theta \) varies from 0 to \( \pi \). This means the described region spans out around the entire upper half of the sphere horizontally, bisecting it from the origin to the perimeter of the cap. If you imagine standing at the pole, the azimuthal angle sweeps out from directly in front of you to directly behind in a half-circle.

The azimuthal angle allows you to locate where, horizontally, a point belongs in the spherical region, enabling a complete search through the specified hemispherical cap region in the given problem.