Question

Use spherical coordinates to find the volume of the following solids. A ball of radius \(a>0\)

Short answer

Answer: The volume of the solid ball of radius \(a\) is \(\frac{4}{3}\pi a^3\).

Step-by-step solution

Understand Spherical Coordinates

Spherical coordinates use three variables \((r, \theta, \phi)\), where \(r\) is the distance from the origin, \(\theta\) is the azimuthal angle, and \(\phi\) is the polar angle. In our case, all points inside the ball can be described using spherical coordinates where \(0 \leq r \leq a\), \(0 \leq \theta \leq 2\pi\), and \(0 \leq \phi \leq \pi\). For any solid object in spherical coordinates, the volume element is given by \(dV = r^2 \sin \phi dr d\theta d\phi\).

Set Up the Integral for the Volume

Now, we will set up the triple integral in spherical coordinates to find the volume of the ball. We will integrate the volume element \(dV\) over the whole ball, i.e. over the range of \(r\), \(\theta\), and \(\phi\) as defined earlier. $$ V = \int_{0}^{\pi}\int_{0}^{2\pi}\int_{0}^{a} r^2 \sin \phi dr d\theta d\phi $$

Evaluate the Integral

Now, we will evaluate the triple integral step by step. First, we will find the integral with respect to \(r\): $$ \int_{0}^{a} r^2 dr = \frac{1}{3} r^3 \Big|_0^a = \frac{1}{3}a^3 $$ Next, we will find the integral with respect to \(\theta\): $$ \int_{0}^{2\pi} d\theta = \theta \Big|_0^{2\pi} = 2\pi $$ Finally, we will find the integral with respect to \(\phi\): $$ \int_{0}^{\pi} \sin \phi d\phi = -\cos \phi \Big|_0^{\pi} = 2 $$

Calculate the Volume

Now, we will multiply the results of all three integrals: $$ V = \frac{1}{3}a^3 \cdot 2\pi \cdot 2 = \frac{4}{3}\pi a^3 $$ So, the volume of the ball of radius \(a\) is \(\frac{4}{3}\pi a^3\).

Key concepts

Triple Integral

When it comes to calculating the volume of a three-dimensional object, one of the most powerful mathematical tools at our disposal is the triple integral. Unlike a single integral, which gives the area under a curve, or a double integral, which provides the volume under a surface, a triple integral extends this concept to compute volumes in three-dimensional space.

A triple integral is a repeated integration process where we integrate over three different variables. For instance, in the Cartesian coordinate system, these variables are typically denoted as x, y, and z. The limits of integration for each variable depend on the geometry of the object whose volume we wish to calculate. More simply, the triple integral accumulates volume 'slices' over the region of interest, adding up infinitesimally small volumes to find the total.

To evaluate a triple integral, we solve the integral sequentially for each variable, starting from the innermost integral and working our way out. It's akin to filling a container layer by layer and then aggregating the layers to determine the total volume.

Volume Element

In the context of multivariable calculus and particularly when using triple integrals, the concept of a volume element is critical. The volume element represents an infinitesimally small portion of the volume of a three-dimensional object.

In the Cartesian coordinate system, the volume element is straightforward and is represented as \(dxdydz\), suggesting a tiny rectangular prism with sides \(dx\), \(dy\), and \(dz\). However, in other coordinate systems, such as spherical or cylindrical coordinates, the volume element takes a different form.

Volume Element in Spherical Coordinates

In spherical coordinates, the volume element is \(r^2 \sin \phi dr d\theta d\phi\), reflecting the geometry of the space. This expression accounts for the non-linear spacing of points as they move away from the origin—the 'r' term counts for radial expansion, the \(\sin \phi\) term adjusts for vertical spacing changes and the product of differential components give our elemental volume shape. When we use such coordinate systems, understanding and using the correct volume element is vital for accurately calculating volumes.

Spherical Coordinate System

The spherical coordinate system is one of several methods for describing points in three-dimensional space. Unlike the Cartesian coordinate system, which uses perpendicular axes to denote position, the spherical coordinate system defines a point by its distance from a reference point (the origin), the angle from a reference direction (azimuthal angle), and the angle from a reference plane (polar angle).

The three variables used are radial distance \(r\), azimuthal angle \(\theta\), and polar angle \(\phi\). Here, \(r\) measures how far away the point is from the origin, \(\theta\) measures the angular position around the vertical axis, and \(\phi\) measures the angle with respect to the positive z-axis.

Advantages of Spherical Coordinates

The spherical coordinate system is especially useful for dealing with problems that have inherent spherical symmetry, such as calculating the fields around a sphere or the volume of spherical objects. When employing this system, it's essential to visualize the angles and distances in three-dimensional space to set up integrals correctly. Additionally, converting from spherical to Cartesian coordinates or vice versa might be necessary depending on the problem at hand.