Evaluate Triple Integral With Cylindrical Coordinates Step-by-Step Solution

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Introduction

In the realm of multivariable calculus, evaluating triple integrals is a fundamental skill, particularly when dealing with volumes and other three-dimensional quantities. Often, the direct computation of a triple integral in Cartesian coordinates can be cumbersome, especially if the region of integration possesses certain symmetries or complexities. In such scenarios, transforming to a different coordinate system, such as cylindrical or spherical coordinates, can significantly simplify the evaluation process. In this article, we will delve into the evaluation of the triple integral ∫₀³ ∫₀² ∫₀^(√(4-y²)) dxdydz using cylindrical coordinates. We will first provide a brief overview of cylindrical coordinates and their relationship to Cartesian coordinates. Then, we will meticulously convert the given integral into cylindrical coordinates, carefully considering the limits of integration. Finally, we will step-by-step compute the integral in cylindrical coordinates, highlighting the advantages of this approach and providing a clear, comprehensive solution.

Understanding Cylindrical Coordinates

Cylindrical coordinates provide an alternative way to represent points in three-dimensional space. Instead of using the Cartesian coordinates (x, y, z), cylindrical coordinates use (r, θ, z), where:

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

The relationships between Cartesian and cylindrical coordinates are given by:

  • x = r cos θ
  • y = r sin θ
  • z = z
  • r² = x² + y²
  • tan θ = y/x

Why use cylindrical coordinates? They are particularly useful when dealing with regions that have cylindrical symmetry, meaning they are symmetric about an axis. In such cases, the integral often simplifies significantly when expressed in cylindrical coordinates. This is because the volume element in cylindrical coordinates, dV, is given by r dz dr dθ, which incorporates the radial distance 'r' and can naturally handle circular or cylindrical shapes. When our region of integration involves cylinders, cones, or portions of spheres centered along the z-axis, transforming to cylindrical coordinates is a strategic move that can drastically reduce the complexity of the integral.

Transforming the Integral to Cylindrical Coordinates

To transform the given integral ∫₀³ ∫₀² ∫₀^(√(4-y²)) dx dy dz into cylindrical coordinates, we need to consider both the integrand and the limits of integration. Let's break this down step by step.

First, let's visualize the region of integration. The limits of integration tell us:

  • z varies from 0 to 3.
  • y varies from 0 to 2.
  • x varies from 0 to √(4 - y²).

The equation x = √(4 - y²) can be rewritten as x² = 4 - y², or x² + y² = 4, with x ≥ 0. This represents the right half of a cylinder of radius 2 centered along the z-axis. Since y varies from 0 to 2, we are considering the first quadrant portion of this half-cylinder.

Now, let's convert the limits of integration to cylindrical coordinates:

  • z: The z-limits remain the same since z is the same in both coordinate systems: 0 ≤ z ≤ 3.
  • y and x: The equation x² + y² = 4 becomes r² = 4, so r = 2 (since r is non-negative). The fact that x varies from 0 to √(4 - y²) and y varies from 0 to 2 means we are in the first quadrant of the circle x² + y² = 4. Thus, θ varies from 0 to Ï€/2.
  • r: Since x² + y² ≤ 4 and we are considering the region where x ≥ 0 and y ≥ 0, the radial distance r varies from 0 to 2.

The differential element dxdydz transforms to r dz dr dθ in cylindrical coordinates. This is a crucial step in the transformation, as it accounts for the change in volume element when switching coordinate systems. Therefore, the transformed integral in cylindrical coordinates is:

∫₀^(π/2) ∫₀² ∫₀³ r dz dr dθ

This transformed integral is now much simpler to evaluate. The cylindrical coordinates have naturally captured the shape of the region, allowing us to set up the integral in a way that makes the computation straightforward. The limits of integration are constants, and the integrand 'r' is a simple function of the radial coordinate. This transformation to cylindrical coordinates is a powerful technique for simplifying integrals over regions with cylindrical symmetry.

Evaluating the Integral in Cylindrical Coordinates

Now that we have transformed the integral into cylindrical coordinates, we can proceed with the evaluation. The integral we have is:

∫₀^(π/2) ∫₀² ∫₀³ r dz dr dθ

We will evaluate this iterated integral from the inside out. First, we integrate with respect to z:

∫₀³ r dz = r [z]₀³ = r(3 - 0) = 3r

Now, we substitute this result back into the integral:

∫₀^(π/2) ∫₀² 3r dr dθ

Next, we integrate with respect to r:

∫₀² 3r dr = 3 [r²/2]₀² = 3(2²/2 - 0²/2) = 3(4/2) = 6

Substituting this back into the integral, we have:

∫₀^(π/2) 6 dθ

Finally, we integrate with respect to θ:

∫₀^(π/2) 6 dθ = 6 [θ]₀^(π/2) = 6(π/2 - 0) = 3π

Therefore, the value of the integral ∫₀³ ∫₀² ∫₀^(√(4-y²)) dxdydz in cylindrical coordinates is 3π. This step-by-step evaluation demonstrates how transforming to cylindrical coordinates can simplify a complex integral into a series of straightforward integrations.

Conclusion

In conclusion, we have successfully evaluated the triple integral ∫₀³ ∫₀² ∫₀^(√(4-y²)) dxdydz using cylindrical coordinates. By recognizing the cylindrical symmetry of the region of integration, we were able to transform the integral into a simpler form that was easily evaluated. The transformation to cylindrical coordinates involved converting the limits of integration and the differential element dxdydz to r dz dr dθ. We then performed the integration iteratively, first with respect to z, then r, and finally θ, to obtain the result 3π.

This example highlights the power of choosing the right coordinate system when evaluating multivariable integrals. Cylindrical coordinates, in particular, are a valuable tool for handling regions with cylindrical symmetry. By mastering this technique, we can tackle a wider range of integration problems and gain a deeper understanding of multivariable calculus.

In summary, the key steps involved in evaluating a triple integral in cylindrical coordinates are:

  1. Visualize the region of integration.
  2. Convert the Cartesian limits of integration to cylindrical coordinates.
  3. Transform the differential element dxdydz to r dz dr dθ.
  4. Evaluate the resulting iterated integral.

By following these steps, we can efficiently and accurately evaluate triple integrals over regions with cylindrical symmetry, unlocking a powerful tool in our mathematical arsenal.