Question

Use spherical coordinates to find the volume of the following solids. The solid cardioid of revolution \(D=\\{(\rho, \varphi, \theta): 0 \leq \rho \leq 1+\cos \varphi, 0 \leq \varphi \leq \pi, 0 \leq \theta \leq 2 \pi\\}\)

Short answer

Question: Find the volume of a solid cardioid of revolution using spherical coordinates. Answer: The volume of the solid cardioid of revolution is \(V = \frac{32\pi^2}{3}\).

Step-by-step solution

1. Calculate the differential volume in spherical coordinates

The differential volume in spherical coordinates is given by \({\rm d}V = \rho^2 \sin\varphi {\rm d}\rho{\rm d}\varphi{\rm d}\theta\). We will use this formula to calculate the volume of the solid by integrating over the given bounds.

2. Set up the triple integral for volume

Using the given bounds for \(\rho\), \(\varphi\), and \(\theta\), we can set up the triple integral for the volume as: $$V = \int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{1+\cos\varphi} \rho^2 \sin\varphi\, {\rm d}\rho{\rm d}\varphi{\rm d}\theta$$

3. Integrate with respect to \(\rho\)

First, we integrate with respect to \(\rho\): $$V = \int_{0}^{2\pi}\int_{0}^{\pi} [\frac{1}{3}\rho^3 \sin\varphi]_{0}^{1+\cos\varphi}\, {\rm d}\varphi{\rm d}\theta$$ On evaluating the integral, we get: $$V = \int_{0}^{2\pi}\int_{0}^{\pi} \frac{1}{3}(1+\cos\varphi)^3 \sin\varphi\, {\rm d}\varphi{\rm d}\theta$$

4. Integrate with respect to \(\varphi\)

Next, integrate with respect to \(\varphi\): $$V = \int_{0}^{2\pi} [\frac{1}{3}(2\cos(\varphi)+1)(3\sin(\varphi)+3\cos(\varphi))]_{0}^{\pi}\,{\rm d}\theta$$ On evaluating the integral, we get: $$V = \int_{0}^{2\pi} \frac{16\pi}{3}\, {\rm d}\theta$$

5. Integrate with respect to \(\theta\)

Finally, integrate with respect to \(\theta\): $$V = [ \frac{16\pi}{3}\theta]_{0}^{2\pi}$$ On evaluating the integral, we get: $$V = \frac{32\pi^2}{3}$$ So, the volume of the solid cardioid of revolution is \(V = \frac{32\pi^2}{3}\).

Key concepts

Triple Integral for Volume

Calculating the volume of a solid using triple integrals is a fundamental concept in multivariable calculus. It involves summing up infinite infinitesimally small 'pieces' or volumes of the solid to find the total volume.

When using rectangular coordinates, these pieces are typically rectangular prisms, but in spherical coordinates, they take the shape of small 'wedges' that stack together to form the solid. The triple integral for volume, in essence, uses a mathematical function that describes how the density (or 'thickness') of the solid changes at different points within the region of integration.

By integrating this density function over the entire volume of the solid, we obtain the total volume. This method is particularly useful for solids with symmetrical properties concerning the origin or for complex shapes where other methods of volume calculation might not be practical.

Cardioid of Revolution

A cardioid is a heart-shaped curve, which can be described using polar or spherical coordinates. It is defined by the equation \(r = 1 + \text{cos}(\varphi)\) in polar coordinates. When this curve is revolved around a specific axis (such as the x-axis in the Cartesian coordinate system), it creates a volume of revolution known as the solid cardioid of revolution.

Understanding the cardioid's shape is crucial for setting up the bounds of integration correctly when computing the volume using triple integrals. For the solid cardioid, you will typically have bounds on \(\rho\), \(\varphi\), and \(\theta\) that delineate the region filled by the volume in spherical coordinates. With these bounds in place, you can use integration to 'add up' all the tiny volumes within the boundaries to find the total volume of the object.

Differential Volume in Spherical Coordinates

In spherical coordinates, the differential volume element, or the 'small piece' of volume we consider when setting up our integrals, is quite different from the rectangular prisms we might be familiar with in Cartesian coordinates.

Due to the geometry of spherical coordinates, this small volume can be imagined as a wedge-shaped 'slice' of a sphere. The differential volume in spherical coordinates is represented by \( \rho^2 \sin(\varphi) \text{d}\rho \text{d}\varphi \text{d}\theta \). Here, \(\rho\) is the radial distance from the origin, \(\varphi\) (also sometimes denoted as \(\theta\)) is the polar angle measured from the positive z-axis, and \(\theta\) (or \(\phi\)) is the azimuthal angle measured from the positive x-axis in the xy-plane.

The expression \(\rho^2 \sin(\varphi)\) comes from the Jacobian determinant, which is necessary when changing variables from Cartesian to spherical coordinates. This term accounts for the 'expansion' and 'contraction' of the volume elements as one moves away from or towards the origin or moves from the equator toward the poles.

Integration in Spherical Coordinates

To integrate in spherical coordinates, one must be thoughtful of the bounds for each variable: \(\rho\), \(\varphi\), and \(\theta\). In the example of the solid cardioid of revolution, the bounds for \(\rho\) are between zero and the equation for the cardioid, for \(\varphi\) between zero and \(\pi\), and for \(\theta\) between zero and \(2\pi\).

In practice, integrating involves three sequential steps: first, you integrate with respect to \(\rho\), then \(\varphi\), and finally \(\theta\). You start from the innermost integral and work your way outwards. Each integration step considers the variable's bounds and the appropriate differential volume element.

The nature of the solid's symmetry often dictates the simplifications that can occur during the integration process. For the solid cardioid, the integrals with respect to \(\varphi\), and \(\theta\) simplify greatly due to the shape's symmetry. By integrating step by step and evaluating at the bounds, the volume of complex three-dimensional shapes like the cardioid of revolution can be computed efficiently and accurately.