Evaluate The Following Integral In Cylindrical Coordinates

Here's how to effectively evaluate integrals using cylindrical coordinates, covering the necessary transformations, practical steps, and illustrative examples.

Evaluating Integrals in Cylindrical Coordinates

Cylindrical coordinates provide a powerful tool for simplifying and solving triple integrals, especially when dealing with regions that possess symmetry about an axis. This coordinate system extends the familiar polar coordinates in the xy-plane to three dimensions, making it particularly useful for problems involving cylinders, cones, and other related shapes.

Understanding Cylindrical Coordinates

Cylindrical coordinates (r, θ, z) represent a point in space using the following parameters:

• r: The distance from the point to the z-axis (the radius). This is the same r as in polar coordinates.
• θ: The angle between the projection of the point onto the xy-plane and the positive x-axis, measured counterclockwise. This is also the same θ as in polar coordinates.
• z: The directed distance from the point to the xy-plane (the height). This is the same z as in Cartesian coordinates.

Relationships with Cartesian Coordinates:

The conversion between cylindrical and Cartesian coordinates (x, y, z) is defined by:

• x = r cos θ
• y = r sin θ
• z = z

Conversely:

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

The Volume Element:

A crucial aspect of using cylindrical coordinates in integration is understanding how the volume element dV transforms. In Cartesian coordinates, dV = dx dy dz. In cylindrical coordinates, it becomes:

• dV = r dz dr dθ

The factor of r arises from the Jacobian determinant of the transformation, which accounts for the scaling effect of the coordinate change. It's essential to include this r in the integrand when setting up the integral in cylindrical coordinates.

When to Use Cylindrical Coordinates

Cylindrical coordinates are most advantageous when the region of integration, or the integrand itself, contains expressions like x² + y². This is because x² + y² simplifies to r² in cylindrical coordinates, often leading to a simpler integral. Here are common scenarios:

• Integrals over Cylinders: If the region of integration is a cylinder or a portion of a cylinder, cylindrical coordinates are almost always the preferred choice.
• Integrals over Cones: Cones are also naturally described using cylindrical coordinates, especially if the cone's axis of symmetry coincides with the z-axis.
• Integrals with Circular Symmetry: When the integrand is a function of x² + y², converting to cylindrical coordinates can greatly simplify the integration process.
• Regions Bounded by Circular Paraboloids: Similar to cones, these surfaces benefit from the r² simplification.

Steps for Evaluating Integrals in Cylindrical Coordinates

Evaluating a triple integral in cylindrical coordinates involves the following key steps:

1. Understand the Region of Integration: The most important first step is to visualize or sketch the region E over which you are integrating. This will help determine the limits of integration for r, θ, and z. Pay close attention to the boundaries of the region. Are they defined by cylinders, planes, cones, or other surfaces?

2. Convert the Integrand to Cylindrical Coordinates: Replace x with r cos θ, y with r sin θ, and leave z as it is in the integrand f(x, y, z). This gives you a new integrand f(r cos θ, r sin θ, z).

3. Determine the Limits of Integration: This is often the most challenging part. You need to express the boundaries of the region E in terms of r, θ, and z.

   • θ Limits: Determine the range of angles θ that sweep out the projection of E onto the xy-plane. This often involves finding the intersection of surfaces and projecting them onto the xy-plane. The limits will be constants if the region has full rotational symmetry.
   • r Limits: For a fixed angle θ, determine the range of radii r that extend from the z-axis to the boundary of the region's projection onto the xy-plane. This involves expressing the boundary curves in polar coordinates (i.e., r as a function of θ). The inner limit might be 0 (the z-axis) or some function of θ.
   • z Limits: For a fixed r and θ, determine the range of z values that span the region E. This involves finding the lower and upper surfaces that bound the region and expressing them as functions of r and θ. The limits will be z = g₁(r, θ) (lower surface) and z = g₂(r, θ) (upper surface).

4. Set Up the Integral: Write the triple integral in cylindrical coordinates as follows:

   ∫∫∫E f(x, y, z) dV = ∫θ₁^{θ₂}_ ∫r₁({θ})_^{r₂}({θ})* ∫g₁({r, θ}*)^{g₂}({r, θ})_ f(r cos θ, r sin θ, z) r dz dr dθ

   Be absolutely sure to include the r in the volume element r dz dr dθ.

5. Evaluate the Integral: Evaluate the integral iteratively, starting with the innermost integral (with respect to z), then the integral with respect to r, and finally the integral with respect to θ. Remember to treat r and θ as constants when integrating with respect to z, and treat θ as a constant when integrating with respect to r.

Example 1: Volume of a Cylinder

Let's find the volume of a cylinder with radius R and height H. The cylinder is defined by x² + y² ≤ R² and 0 ≤ z ≤ H.

1. Region of Integration: The region is a cylinder.

2. Integrand: Since we're finding the volume, the integrand is simply f(x, y, z) = 1. In cylindrical coordinates, this remains f(r cos θ, r sin θ, z) = 1.

3. Limits of Integration:

   • θ: The cylinder spans the entire circle, so 0 ≤ θ ≤ 2π.
   • r: The radius ranges from the z-axis to the edge of the cylinder, so 0 ≤ r ≤ R.
   • z: The height ranges from the base to the top of the cylinder, so 0 ≤ z ≤ H.

4. Set Up the Integral:

   ∫∫∫E dV = ∫0^{2π}_ ∫0^{R}_ ∫0^{H}_ r dz dr dθ

5. Evaluate the Integral:

   ∫0^{2π}_ ∫0^{R}_ ∫0^{H}_ r dz dr dθ = ∫0^{2π}_ ∫0^{R}_ [rz]0^H dr dθ = ∫0^*{2π}* ∫0^{R}_ Hr dr dθ = ∫0^{2π}_ [H(r²/2)]0^R dθ = ∫0^*{2π}* (HR²/2) dθ = [* (HR²/2)θ*]_0^{2π} = πR²H

This result confirms the well-known formula for the volume of a cylinder.

Example 2: Volume of a Region Bounded by a Paraboloid and a Plane

Find the volume of the solid E bounded by the paraboloid z = 16 - x² - y² and the plane z = 0.

1. Region of Integration: The region is bounded below by the xy-plane and above by the paraboloid. The paraboloid intersects the xy-plane when 16 - x² - y² = 0, which implies x² + y² = 16. This is a circle of radius 4 in the xy-plane.

2. Integrand: Again, we're finding the volume, so f(x, y, z) = 1.

3. Limits of Integration:

   • θ: The region projects onto the entire circle x² + y² ≤ 16, so 0 ≤ θ ≤ 2π.
   • r: The radius ranges from the z-axis to the circle x² + y² = 16, so 0 ≤ r ≤ 4.
   • z: The height ranges from the plane z = 0 to the paraboloid z = 16 - x² - y² = 16 - r², so 0 ≤ z ≤ 16 - r².

4. Set Up the Integral:

   ∫∫∫E dV = ∫0^{2π}_ ∫0^{4}_ ∫0^{16-r²}_ r dz dr dθ

5. Evaluate the Integral:

   ∫0^{2π}_ ∫0^{4}_ ∫0^{16-r²}_ r dz dr dθ = ∫0^{2π}_ ∫0^{4}_ [rz]0^{16-r²} dr dθ = ∫0^*{2π}* ∫0^{4}_ r(16 - r²) dr dθ = ∫0^{2π}_ ∫0^{4}_ (16r - r³) dr dθ = ∫0^{2π}_ [8r² - (r⁴/4)]0^4 dθ = ∫0^*{2π}* (8(16) - (256/4)) dθ = ∫0^{2π}_ (128 - 64) dθ = ∫0^{2π}_ 64 dθ = [64θ]_0^{2π} = 128π

Therefore, the volume of the solid is 128π.

Example 3: Evaluating a Density Integral

Suppose a solid occupies the region defined by x² + y² ≤ 9 and 0 ≤ z ≤ 5, and its density is given by ρ(x, y, z) = x² + y². Find the mass of the solid.

1. Region of Integration: The region is a cylinder with radius 3 and height 5.

2. Integrand: The density function is ρ(x, y, z) = x² + y². In cylindrical coordinates, this becomes ρ(r cos θ, r sin θ, z) = r². Since mass is the integral of density over volume, the integrand for the mass integral is ρ(r cos θ, r sin θ, z) = r².

3. Limits of Integration:

   • θ: The cylinder spans the entire circle, so 0 ≤ θ ≤ 2π.
   • r: The radius ranges from the z-axis to the edge of the cylinder, so 0 ≤ r ≤ 3.
   • z: The height ranges from the base to the top of the cylinder, so 0 ≤ z ≤ 5.

4. Set Up the Integral:

   Mass = ∫∫∫E* ρ(x, y, z) dV = ∫0^{2π}_ ∫0^{3}_ ∫0^{5}_ r² * r dz dr dθ = ∫0^{2π}_ ∫0^{3}_ ∫0^{5}_ r³ dz dr dθ

5. Evaluate the Integral:

   ∫0^{2π}_ ∫0^{3}_ ∫0^{5}_ r³ dz dr dθ = ∫0^{2π}_ ∫0^{3}_ [r³z]0^5 dr dθ = ∫0^*{2π}* ∫0^{3}_ 5r³ dr dθ = ∫0^{2π}_ [5(r⁴/4)]0^3 dθ = ∫0^*{2π}* (5(81)/4) dθ = ∫0^{2π}_ (405/4) dθ = [(405/4)θ]_0^{2π} = (405/2)π

The mass of the solid is (405/2)π.

Common Mistakes to Avoid

• Forgetting the Jacobian: The most common mistake is forgetting to include the r in the volume element dV = r dz dr dθ. This will lead to an incorrect result.
• Incorrect Limits of Integration: Carefully determine the limits of integration based on the geometry of the region. Sketching the region is highly recommended.
• Confusing r and θ: Make sure you understand the roles of r and θ in defining the region. r is the radial distance, and θ is the angle.
• Incorrectly Converting the Integrand: Double-check that you have correctly substituted x = r cos θ and y = r sin θ into the integrand.
• Order of Integration: While the order of integration can sometimes be changed, it's important to choose an order that simplifies the integral as much as possible. Integrating with respect to z first is often a good strategy.

Practical Tips for Success

• Sketch the Region: Always start by sketching the region of integration. This will help you visualize the limits of integration and identify any symmetries that can simplify the problem.
• Start with the Easiest Limits: Often, the limits for θ are the easiest to determine. Start with those.
• Work from the Inside Out: Evaluate the integral iteratively, starting with the innermost integral.
• Check Your Answer: If possible, use geometric intuition or other methods to check if your answer is reasonable.
• Practice, Practice, Practice: The best way to master cylindrical coordinates is to work through a variety of examples.

Advanced Considerations

• Symmetry: Exploit symmetry whenever possible. If the region and the integrand are symmetric with respect to the z-axis, you may be able to reduce the integration range for θ. If the region is symmetric about the xy-plane and the integrand is odd with respect to z, the integral will be zero.
• Changing the Order of Integration: While dz dr dθ is a common order, you can sometimes simplify the integral by changing the order of integration. However, this requires careful consideration of the limits of integration.
• Using Computer Algebra Systems (CAS): For complex integrals, consider using a CAS such as Mathematica, Maple, or SymPy to help with the evaluation. These tools can handle symbolic integration and provide accurate numerical results.

Conclusion

Evaluating integrals in cylindrical coordinates is a powerful technique for solving problems involving regions with axial symmetry. By understanding the coordinate system, following the steps outlined above, and avoiding common mistakes, you can effectively tackle a wide range of triple integral problems. Remember to practice regularly and use visualization to aid in your understanding. The ability to skillfully apply cylindrical coordinates will significantly enhance your problem-solving capabilities in calculus and related fields. The key is to visualize the region, correctly determine the limits of integration, and carefully perform the integration steps.