Transform To Polar Coordinates And Evaluate Integral

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#Introduction

In the realm of multivariable calculus, double integrals play a crucial role in determining various quantities such as area, volume, and mass. Often, the direct evaluation of a double integral in Cartesian coordinates can be challenging due to the complexity of the region of integration or the integrand itself. In such cases, a transformation to a more suitable coordinate system, such as polar coordinates, can significantly simplify the problem. This article delves into the process of transforming a double integral from Cartesian to polar coordinates and demonstrates its application in evaluating a specific integral. We will explore the underlying theory, the steps involved in the transformation, and the subsequent evaluation, providing a comprehensive understanding of this powerful technique.

Understanding Polar Coordinates

Before we embark on the transformation process, it is essential to have a firm grasp of polar coordinates. Unlike the Cartesian coordinate system, which uses two perpendicular axes (x and y) to define a point in the plane, polar coordinates employ a radial distance (r) and an angle (θ) to locate a point. The radial distance, r, represents the distance from the origin to the point, while the angle, θ, measures the counterclockwise rotation from the positive x-axis to the line segment connecting the origin and the point. This alternative representation offers a more natural way to describe certain geometric shapes and regions, particularly those with circular symmetry.

The relationship between Cartesian coordinates (x, y) and polar coordinates (r, θ) is defined by the following equations:

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

These equations allow us to seamlessly convert between the two coordinate systems, enabling us to choose the most appropriate system for a given problem. In the context of double integrals, the transformation to polar coordinates can simplify the integrand and the region of integration, making the evaluation process more manageable. For instance, regions bounded by circles or circular arcs are often more easily described in polar coordinates, as the radial distance remains constant along these boundaries. Similarly, integrands involving expressions like x² + y² can be simplified using the identity r² = x² + y², leading to a more tractable integral.

Transforming Double Integrals to Polar Coordinates

The transformation of a double integral from Cartesian to polar coordinates involves several key steps. First, we must identify the region of integration in Cartesian coordinates and determine its equivalent representation in polar coordinates. This involves expressing the boundaries of the region in terms of r and θ. Next, we need to rewrite the integrand in terms of r and θ using the transformation equations mentioned earlier. Finally, we must account for the change in area element when transforming from Cartesian to polar coordinates. In Cartesian coordinates, the area element is given by dA = dx dy, while in polar coordinates, the area element is given by dA = r dr dθ. The extra factor of 'r' arises from the fact that the area of a small polar rectangle is approximately r Δr Δθ, where Δr and Δθ are small increments in r and θ, respectively.

Therefore, the general transformation formula for a double integral from Cartesian to polar coordinates is given by:

∬R f(x, y) dA = ∬R' f(r cos θ, r sin θ) r dr dθ

where R is the region of integration in Cartesian coordinates and R' is the corresponding region in polar coordinates. This formula encapsulates the essence of the transformation, highlighting the importance of expressing the integrand and the area element in terms of polar coordinates.

Step-by-Step Transformation Process

Let's outline the step-by-step process of transforming a double integral to polar coordinates:

  1. Sketch the region of integration: This step is crucial for visualizing the region and determining its boundaries. Plot the curves that define the region in the Cartesian plane.
  2. Determine the limits of integration in polar coordinates: Express the boundaries of the region in terms of r and θ. This involves finding the minimum and maximum values of r and θ that cover the region.
  3. Rewrite the integrand in polar coordinates: Substitute x = r cos θ and y = r sin θ into the integrand.
  4. Replace the area element: Replace dA with r dr dθ.
  5. Evaluate the integral: Evaluate the resulting double integral in polar coordinates using appropriate integration techniques.

By following these steps systematically, we can effectively transform a double integral to polar coordinates and simplify the evaluation process.

Evaluating the Integral: A Step-by-Step Solution

Now, let's apply the transformation technique to evaluate the given double integral:

∫0a ∫0√(a² - y²) y√(x² + y²) dx dy

1. Sketching the Region of Integration

The region of integration is defined by the following inequalities:

  • 0 ≤ y ≤ a
  • 0 ≤ x ≤ √(a² - y²)

The inequality 0 ≤ y ≤ a indicates that the region lies above the x-axis and below the horizontal line y = a. The inequality 0 ≤ x ≤ √(a² - y²) represents the right half of the circle x² + y² = a². Combining these inequalities, we find that the region of integration is the quarter-circle in the first quadrant with radius a, centered at the origin.

2. Determining the Limits of Integration in Polar Coordinates

In polar coordinates, the quarter-circle in the first quadrant can be described by the following inequalities:

  • 0 ≤ r ≤ a
  • 0 ≤ θ ≤ π/2

These inequalities indicate that the radial distance r varies from 0 to a, and the angle θ varies from 0 to π/2, covering the entire quarter-circle.

3. Rewriting the Integrand in Polar Coordinates

Substituting x = r cos θ and y = r sin θ into the integrand, we get:

y√(x² + y²) = (r sin θ)√(r² cos² θ + r² sin² θ) = (r sin θ)√(r²(cos² θ + sin² θ)) = (r sin θ)√r² = r² sin θ

Thus, the integrand in polar coordinates is r² sin θ.

4. Replacing the Area Element

Replacing the area element dA with r dr dθ, we obtain the transformed integral:

∫0π/2 ∫0a (r² sin θ) r dr dθ = ∫0π/2 ∫0a r³ sin θ dr dθ

5. Evaluating the Integral

Now, we can evaluate the double integral in polar coordinates. First, we integrate with respect to r:

∫0π/2 [∫0a r³ sin θ dr] dθ = ∫0π/2 [sin θ ∫0a r³ dr] dθ = ∫0π/2 [sin θ (r⁴/4)|0a] dθ = ∫0π/2 (a⁴/4) sin θ dθ

Next, we integrate with respect to θ:

(a⁴/4) ∫0π/2 sin θ dθ = (a⁴/4) (-cos θ)|0π/2 = (a⁴/4) (-cos(π/2) + cos(0)) = (a⁴/4) (0 + 1) = a⁴/4

Therefore, the value of the double integral is a⁴/4.

Conclusion

In this article, we have explored the technique of transforming double integrals from Cartesian to polar coordinates. We have seen how this transformation can simplify the evaluation of integrals by adapting the coordinate system to the geometry of the region of integration and the form of the integrand. By understanding the relationship between Cartesian and polar coordinates and following a systematic transformation process, we can effectively tackle a wide range of double integral problems. The example we discussed demonstrates the power of this technique, allowing us to evaluate a challenging integral with relative ease. The transformation to polar coordinates is a valuable tool in the arsenal of any mathematician or scientist dealing with multivariable calculus, offering a powerful approach to solving problems involving integrals over regions with circular symmetry or integrands that simplify in polar form. This technique is not just a mathematical trick but a fundamental concept that highlights the importance of choosing the right coordinate system to simplify a problem, a principle that extends far beyond the realm of calculus and into various areas of science and engineering.