Triple Integral Of A Sphere In Cylindrical Coordinates

The triple integral is a powerful tool for calculating volumes, masses, and other properties of three-dimensional objects. When dealing with spheres, cylindrical coordinates offer a convenient way to simplify the integration process, particularly when the sphere is centered at the origin or has an axis of symmetry aligned with the z-axis. This article will guide you through the process of setting up and evaluating a triple integral for a sphere using cylindrical coordinates, covering the theoretical foundations, practical steps, and common challenges.

Understanding Cylindrical Coordinates

Cylindrical coordinates are a three-dimensional coordinate system that extends the two-dimensional polar coordinate system. Instead of using Cartesian coordinates (x, y, z), cylindrical coordinates use (r, θ, z), where:

• r is the distance from the point to the z-axis (the radius in the xy-plane).
• θ is the angle between the positive x-axis and the projection of the point onto the xy-plane.
• z is the same as in Cartesian coordinates, representing the height above the xy-plane.

The relationships between Cartesian and cylindrical coordinates are:

• x = r cos θ
• y = r sin θ
• z = z

And conversely:

• r = √(x² + y²)
• θ = arctan(y/x)
• z = z

When transforming a triple integral from Cartesian to cylindrical coordinates, we also need to account for the Jacobian determinant, which represents the scaling factor due to the coordinate transformation. For cylindrical coordinates, the Jacobian is r. This means that the volume element dV in Cartesian coordinates (dx dy dz) becomes r dr dθ dz in cylindrical coordinates.

Setting Up the Triple Integral for a Sphere

Consider a sphere with radius a centered at the origin. Its equation in Cartesian coordinates is:

x² + y² + z² = a²

To set up the triple integral in cylindrical coordinates, we need to express this equation in terms of r, θ, and z. Substituting x = r cos θ and y = r sin θ, we get:

(r cos θ)² + (r sin θ)² + z² = a²

r²(cos² θ + sin² θ) + z² = a²

Since cos² θ + sin² θ = 1, the equation simplifies to:

r² + z² = a²

Now, we need to determine the limits of integration for r, θ, and z.

Limits of Integration

1. θ (Angle): For a complete sphere, the angle θ ranges from 0 to 2π, covering all angles in the xy-plane.

   0 ≤ θ ≤ 2π

2. r (Radius): For a given z, the radius r ranges from 0 to the radius of the circle formed by the intersection of the sphere and the plane at height z. From the equation r² + z² = a², we can solve for r:

   r² = a² - z²

   r = √(a² - z²)

   So, the limits for r are:

   0 ≤ r ≤ √(a² - z²)

3. z (Height): The height z ranges from the bottom to the top of the sphere. From the equation r² + z² = a², when r = 0 (at the poles of the sphere), we have z² = a², which gives us z = ±a. Thus, the limits for z are:

   -a ≤ z ≤ a

The Triple Integral

With the limits of integration determined, the triple integral for calculating the volume V of the sphere is:

V = ∫∫∫ dV = ∫₀²π ∫₋ₐᵃ ∫₀^(√(a²-z²)) r dr dz dθ

Note the order of integration. While the order can be changed, this order is logical because the radius r is dependent on z, and both are independent of θ.

Evaluating the Triple Integral

Now, let's evaluate the triple integral step-by-step.

1. Integrate with respect to r:

   ∫₀^(√(a²-z²)) r dr = [r²/2]₀^(√(a²-z²)) = (a² - z²)/2

2. Integrate with respect to z:

   ∫₋ₐᵃ (a² - z²)/2 dz = (1/2) ∫₋ₐᵃ (a² - z²) dz = (1/2) [a²z - z³/3]₋ₐᵃ

   = (1/2) [(a³ - a³/3) - (-a³ + a³/3)] = (1/2) [2a³ - 2a³/3] = (1/2) [4a³/3] = (2a³)/3

3. Integrate with respect to θ:

   ∫₀²π (2a³)/3 dθ = (2a³/3) ∫₀²π dθ = (2a³/3) [θ]₀²π = (2a³/3) (2π) = (4πa³)/3

Therefore, the volume of the sphere with radius a is (4πa³)/3.

Alternative Approach: Exploiting Symmetry

Sometimes, exploiting symmetry can simplify the integral. For instance, we can calculate the volume of the upper hemisphere (0 ≤ z ≤ a) and then multiply the result by 2. In this case, the limits of integration would be:

• 0 ≤ θ ≤ 2π
• 0 ≤ r ≤ √(a² - z²)
• 0 ≤ z ≤ a

The triple integral for the upper hemisphere would be:

V_half = ∫₀²π ∫₀ᵃ ∫₀^(√(a²-z²)) r dr dz dθ

Evaluating this integral:

1. Integrate with respect to r:

   ∫₀^(√(a²-z²)) r dr = [r²/2]₀^(√(a²-z²)) = (a² - z²)/2

2. Integrate with respect to z:

   ∫₀ᵃ (a² - z²)/2 dz = (1/2) ∫₀ᵃ (a² - z²) dz = (1/2) [a²z - z³/3]₀ᵃ = (1/2) [a³ - a³/3] = (1/2) [2a³/3] = (a³)/3

3. Integrate with respect to θ:

   ∫₀²π (a³)/3 dθ = (a³/3) ∫₀²π dθ = (a³/3) [θ]₀²π = (a³/3) (2π) = (2πa³)/3

Multiplying by 2 to get the volume of the whole sphere:

V = 2 * (2πa³)/3 = (4πa³)/3

This method provides the same result but might be easier to compute for some individuals.

Practical Considerations and Challenges

While cylindrical coordinates simplify the integration of spheres, there are some practical considerations and challenges to keep in mind:

• Orientation of the Sphere: The simplification is greatest when the sphere is centered at the origin. If the sphere is shifted or rotated, the limits of integration become more complex. You may need to perform a coordinate transformation to align the sphere with the z-axis.
• Complexity of the Integrand: If the integrand (the function being integrated) is more complex than just 1 (for volume), the integration process can become significantly more challenging, even in cylindrical coordinates.
• Choice of Coordinate System: While cylindrical coordinates are often a good choice for spheres, spherical coordinates may be even more suitable in certain cases, especially if the integrand also exhibits spherical symmetry.
• Visualizing the Region of Integration: Accurately visualizing the three-dimensional region of integration is crucial for determining the correct limits. Sketching the sphere and its projections onto the coordinate planes can be helpful.
• Software Assistance: For complex integrals, using computer algebra systems (CAS) like Mathematica, Maple, or Python with SymPy can be invaluable for performing the integration and verifying results.

When to Use Cylindrical Coordinates

Cylindrical coordinates are particularly useful in the following situations:

• Regions with Cylindrical Symmetry: When the region of integration is symmetric around the z-axis, cylindrical coordinates can greatly simplify the integral. This includes not just spheres but also cylinders, cones, and other solids of revolution.
• Integrands Involving x² + y²: If the integrand contains expressions like x² + y², converting to cylindrical coordinates (where x² + y² = r²) can simplify the integrand.
• Problems Involving Circular Boundaries: When the region of integration has circular boundaries in the xy-plane, cylindrical coordinates are a natural choice.
• Setting up integrals for objects defined by a radius and height Situations where you have a clear radius extending from a central axis (z) and varying height above some plane are perfectly tailored for cylindrical coordinates.

Examples Beyond Simple Volume Calculation

The triple integral in cylindrical coordinates can be used for more than just calculating the volume of a sphere. Here are a few examples:

1. Calculating the Mass of a Sphere with Variable Density: Suppose the density of the sphere is not uniform but varies with the distance from the center, ρ(r, θ, z) = k√(x² + y² + z²) = k√(r² + z²), where k is a constant. The mass M of the sphere would be:

   M = ∫∫∫ ρ(r, θ, z) dV = ∫₀²π ∫₋ₐᵃ ∫₀^(√(a²-z²)) k√(r² + z²) * r dr dz dθ

   This integral is more complex but can still be evaluated using appropriate techniques.

2. Finding the Center of Mass of a Hemisphere: To find the z-coordinate of the center of mass (z̄) of a uniform hemisphere, we use the formula:

   z̄ = (1/V) ∫∫∫ z dV

   Where V is the volume of the hemisphere. In cylindrical coordinates:

   z̄ = (3/(2πa³)) ∫₀²π ∫₀ᵃ ∫₀^(√(a²-z²)) z * r dr dz dθ

3. Calculating the Moment of Inertia: The moment of inertia I of a sphere about the z-axis is given by:

   I = ∫∫∫ (x² + y²) dV = ∫∫∫ r² dV = ∫₀²π ∫₋ₐᵃ ∫₀^(√(a²-z²)) r³ dr dz dθ

   This calculation can be used, for instance, to determine how difficult it is to spin a sphere about its vertical axis.

Common Mistakes to Avoid

When working with triple integrals in cylindrical coordinates, be aware of the following common mistakes:

• Forgetting the Jacobian: Always remember to include the Jacobian determinant r when transforming from Cartesian to cylindrical coordinates.
• Incorrect Limits of Integration: Carefully determine the limits of integration for r, θ, and z. Visualizing the region and solving for the limits algebraically is crucial.
• Incorrect Order of Integration: While the order of integration can sometimes be changed, it's important to choose an order that simplifies the integral. Integrating with respect to a variable that depends on other variables should be done first.
• Sign Errors: Be careful with signs when evaluating the limits of integration, especially when dealing with negative values.
• Computational Errors: Triple integrals can be tedious to evaluate by hand. Double-check your calculations and consider using software to verify your results.

FAQ

Q: Why use cylindrical coordinates instead of Cartesian coordinates for a sphere?

A: Cylindrical coordinates exploit the symmetry of the sphere around the z-axis, making the limits of integration simpler. Specifically, they allow you to express the radius r easily in terms of z.

Q: Can I use cylindrical coordinates for any sphere, regardless of its position?

A: Yes, but the simplification is greatest when the sphere is centered at the origin or has an axis of symmetry aligned with the z-axis. For spheres in other positions, you may need to perform a coordinate translation first.

Q: Is there a case when spherical coordinates are better than cylindrical coordinates for a sphere?

A: Yes, if the integrand also has spherical symmetry. Spherical coordinates use radius, and two angles, directly reflecting the sphere's natural parameters.

Q: What is the Jacobian and why is it important?

A: The Jacobian is a determinant that accounts for the scaling factor due to coordinate transformations. In cylindrical coordinates, it's r, and it's essential for correctly calculating the volume element dV.

Q: How do I handle a sphere that is not perfectly centered at the origin?

A: You can translate the coordinate system so that the origin is at the center of the sphere. This involves a simple shift in the z-coordinate.

Conclusion

The triple integral in cylindrical coordinates provides a powerful method for calculating various properties of spheres. By understanding the transformation from Cartesian to cylindrical coordinates, carefully determining the limits of integration, and avoiding common mistakes, you can effectively apply this technique to solve a wide range of problems. Remember to visualize the region of integration, exploit symmetry when possible, and consider using software to assist with complex calculations. Mastery of this technique not only enhances your mathematical skills but also provides valuable insights into the physical properties of three-dimensional objects.