Evaluate The Following Integral In Cylindrical Coordinates

Evaluating Integrals in Cylindrical Coordinates: A Comprehensive Guide

Evaluating integrals, especially those involving complex geometries, can be significantly simplified by choosing the right coordinate system. While Cartesian coordinates are familiar, cylindrical coordinates offer a powerful alternative for problems exhibiting cylindrical symmetry. This comprehensive guide delves into the intricacies of evaluating integrals in cylindrical coordinates, providing a step-by-step approach and illustrative examples.

Understanding Cylindrical Coordinates

Before diving into integration, let's solidify our understanding of cylindrical coordinates. Instead of using the familiar (x, y, z) Cartesian system, we represent a point in 3D space using (r, θ, z):

• r: The radial distance from the z-axis. This is always non-negative.
• θ: The azimuthal angle (polar angle) measured counterclockwise from the positive x-axis. This typically ranges from 0 to 2π.
• z: The height or vertical coordinate, identical to the z-coordinate in Cartesian coordinates.

The transformation between Cartesian and cylindrical coordinates is defined by:

• x = r cos θ
• y = r sin θ
• z = z

Conversely:

• r = √(x² + y²)
• θ = arctan(y/x) (Carefully consider the quadrant when using arctan)
• z = z

The Jacobian Determinant: A Crucial Element

The Jacobian determinant is the key to transforming integrals from Cartesian to cylindrical coordinates. It accounts for the scaling effect of the coordinate transformation. For cylindrical coordinates, the Jacobian determinant, denoted as J, is:

J = ∣∂(x, y, z)/∂(r, θ, z)∣ = r

This means that the infinitesimal volume element dV in Cartesian coordinates (dx dy dz) transforms to r dr dθ dz in cylindrical coordinates.

Setting up the Integral in Cylindrical Coordinates

To evaluate a triple integral in cylindrical coordinates, we follow these steps:

1. Identify the Region of Integration: Carefully define the region of integration in terms of r, θ, and z. This often involves analyzing the problem's geometry and boundaries. Sketching the region is highly recommended.

2. Express the Integrand: Rewrite the integrand function, f(x, y, z), in terms of r, θ, and z using the transformation equations mentioned earlier.

3. Set up the Integral: The general form of a triple integral in cylindrical coordinates is:

   ∫∫∫_V f(r, θ, z) *r dr dθ dz

4. Determine the Limits of Integration: Based on the region of integration, determine the limits for r, θ, and z. These limits will be constants or functions of other variables. The order of integration can influence the complexity of the calculation; choose the order that simplifies the integration process.

5. Evaluate the Integral: Proceed with the integration, starting with the innermost integral and working outwards.

Illustrative Examples

Let's solidify our understanding with a few examples:

Example 1: Integrating over a Cylinder

Consider the integral of the function f(x, y, z) = x² + y² over the cylinder defined by x² + y² ≤ 4, 0 ≤ z ≤ 3.

1. Region of Integration: The region is a cylinder with radius 2 and height 3. In cylindrical coordinates: 0 ≤ r ≤ 2, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ 3.

2. Integrand: f(x, y, z) = x² + y² = r²

3. Integral Setup:

   ∫₀³ ∫₀^2π ∫₀² r²(r dr dθ dz)

4. Evaluation:

   = ∫₀³ ∫₀^2π [r⁴/4]₀² dθ dz = ∫₀³ ∫₀^2π 4 dθ dz = ∫₀³ [4θ]₀^2π dz = ∫₀³ 8π dz = [8πz]₀³ = 24π

Example 2: Integrating over a Cone

Let's integrate f(x, y, z) = z over the region inside the cone z = √(x² + y²) and below the plane z = 1.

1. Region of Integration: The region is a cone with apex at the origin and height 1. In cylindrical coordinates: 0 ≤ r ≤ z, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ 1.

2. Integrand: f(x, y, z) = z

3. Integral Setup:

   ∫₀^2π ∫₀¹ ∫₀^z z *r dr dθ dz

4. Evaluation:

   = ∫₀^2π ∫₀¹ z [r²/2]₀^z dθ dz = ∫₀^2π ∫₀¹ z³/2 dθ dz = ∫₀^2π [z⁴/8]₀¹ dθ = ∫₀^2π (1/8) dθ = [θ/8]₀^2π = π/4

Example 3: A More Complex Scenario

Consider integrating f(x, y, z) = x over the region bounded by the paraboloid z = x² + y² and the plane z = 1.

1. Region of Integration: The region is bounded below by the paraboloid and above by the plane. In cylindrical coordinates: 0 ≤ r ≤ √z, 0 ≤ θ ≤ 2π, 0 ≤ z ≤ 1. Note that the radius 'r' depends on 'z' here.

2. Integrand: f(x, y, z) = x = r cos θ

3. Integral Setup:

   ∫₀^2π ∫₀¹ ∫₀^√z (r cos θ) *r dr dθ dz

4. Evaluation:

   = ∫₀^2π ∫₀¹ cos θ [r³/3]₀^√z dz dθ = ∫₀^2π ∫₀¹ (z√z)/3 cos θ dz dθ = ∫₀^2π cos θ [z^5/2/(5/2 * 3)]₀¹ dθ = ∫₀^2π (2/15) cos θ dθ = [2sin θ / 15]₀^2π = 0

These examples demonstrate the power and flexibility of cylindrical coordinates in evaluating integrals. The choice of coordinate system significantly impacts the complexity of the problem. Always carefully consider the symmetry of the region and the nature of the integrand when selecting your coordinate system. Remember that practice is key; tackling various problems will build your proficiency in this crucial technique. Remember to always double-check your limits of integration and your integrand transformation. A well-drawn diagram will greatly assist in avoiding common errors.