Question

Evaluate the integral which is given in cylindrical or spherical coordinates, and describe the region \(R\) of integration. $$ \int_{0}^{\pi} \int_{0}^{2 \pi} \int_{0}^{a} \rho^{2} \sin \phi d \rho d \theta d \phi $$

Short answer

The integral evaluates to \( \frac{4\pi a^3}{3} \), representing the volume of a sphere with radius \( a \).

Step-by-step solution

Understanding the Integral

The given integral is expressed in spherical coordinates. The variables are \( \rho \), \( \theta \), and \( \phi \) which represent the radial distance, the azimuthal angle, and the polar angle, respectively. The integrand is \( \rho^2 \sin \phi \).

Describing the Region of Integration

The region \( R \) of integration corresponds to a spherical region. It is defined by \( 0 \leq \rho \leq a \), \( 0 \leq \theta \leq 2\pi \), and \( 0 \leq \phi \leq \pi \). This represents a full sphere of radius \( a \).

Integrate with Respect to \( \rho \)

Integrate the innermost term with respect to \( \rho \):\[ \int_{0}^{a} \rho^2 \ d\rho = \left[ \frac{\rho^3}{3} \right]_{0}^{a} = \frac{a^3}{3}. \]

Integrate with Respect to \( \theta \)

Next, integrate the resulting expression with respect to \( \theta \) over the interval \([0, 2\pi]\):\[ \int_{0}^{2\pi} \frac{a^3}{3} \ d\theta = \frac{a^3}{3} \times \left[ \theta \right]_{0}^{2\pi} = \frac{a^3}{3} \times (2\pi - 0) = \frac{2\pi a^3}{3}.\]

Integrate with Respect to \( \phi \)

Finally, integrate with respect to \( \phi \) over the interval \([0, \pi]\) while remembering to multiply by \( \sin \phi \):\[ \int_{0}^{\pi} \frac{2\pi a^3}{3} \sin \phi \ d\phi = \frac{2\pi a^3}{3} \times [-\cos \phi]_{0}^{\pi} = \frac{2\pi a^3}{3} \times (1 + 1) = \frac{4\pi a^3}{3}.\]

Final Result

The final value of the integral is \( \frac{4\pi a^3}{3} \), which represents the volume of a sphere with radius \( a \).

Key concepts

Triple Integration

Triple integration is a powerful mathematical technique used to calculate volumes in three-dimensional spaces where limits are defined in terms of three variables. In the exercise, we use triple integration to find the volume of a sphere using spherical coordinates. The three integrals are performed one after another, each considering a different dimension of the problem. Here, the integration proceeds from the innermost variable, \( \rho \), to the outermost variable, \( \phi \).

• The innermost integral \( \int_{0}^{a} \) represents integration over the radial distance, \( \rho \).
• Next, the integral \( \int_{0}^{2 \pi} \) integrates over the azimuthal angle, \( \theta \).
• Finally, the outermost integral \( \int_{0}^{\pi} \) involves the polar angle, \( \phi \).

Each integral builds on the previous one, turning a complex three-dimensional region into a manageable calculation.

Volume of a Sphere

One key outcome of this integral is finding the volume of a sphere with radius \(a\). The expression \( \frac{4\pi a^3}{3} \) appears in the final step and represents the sphere's volume.This formula can be derived from the integral of spherical coordinates, which perfectly encapsulates the volume within the defined spherical region:- After integrating with respect to \( \rho \), the radial component, it provides a piece of the volume based on the radius cubed, or \(a^3\).- Integrating over \( \theta \) accounts for the full rotation around the azimuthal angle.- The integration over \( \phi \) considers the hemisphere, multiplying the hemisphere result by 2 to complete the sphere.This systematic approach elegantly derives the same formula we learn for a sphere's volume in geometry.

Spherical Region

In spherical coordinates, a spherical region is often defined by three limits corresponding to radial, polar, and azimuthal dimensions. These ranges form a full, symmetrical region in three-dimensional space.The limits provided in the integral description, \( 0 \leq \rho \leq a \), \( 0 \leq \theta \leq 2\pi \), and \( 0 \leq \phi \leq \pi \), describe a complete sphere of radius \(a\).

• \( \rho \): Varies from the origin to the sphere's radius \(a\), accounting for all points within the sphere's radial distance.
• \( \theta \): Sweeps through all possible angles, a full rotation, to span the spherical shell horizontally.
• \( \phi \): Sweeps from the positive z-axis down to the negative z-axis, covering the sphere vertically.

This spherical symmetry simplifies the calculation and depicts the entirety of a sphere in terms of mathematical limits.

Polar Coordinates

Polar coordinates are used to define a point's location through a radius and angle in two-dimensional space. In our situation, they are part of the broader spherical coordinates system.In spherical coordinates, the polar aspects emerge in terms of the angles \( \theta \) and \( \phi \):- \( \theta \) is akin to the polar angle, describing a point's rotation around the z-axis. Similar to polar coordinates, it defines the angle in a plane but extends it to 3D.- \( \phi \) further helps in understanding a point's position in 3D by introducing a vertical component.The transition from polar to spherical coordinates essentially allows us to better handle problems involving full three-dimensional region integration like spheres. For students who have dealt with polar coordinates, understanding spherical coordinates becomes a natural extension, as they similarly revolve around angles and distances, but add complexity by incorporating depth.