Question

Use polar coordinates to find the centroid of the following constant-density plane regions. The quarter-circular disk \(R=\\{(r, \theta): 0 \leq r \leq 2,0 \leq \theta \leq \pi / 2\\}\)

Short answer

The centroid coordinates of the quarter-circular plane region R are \((\frac{2}{\pi}, \frac{2}{\pi})\).

Step-by-step solution

Set up the Integral for the Area

First, we need to find the area A of the quarter-circle. To do this, we'll set up an integral in polar coordinates. The area of a polar curve can be found using the equation: $$A = \frac{1}{2} \int_{\alpha}^{\beta} r^2 d\theta$$ where α and β are the limits of integration for the θ coordinate. In this case, \(\alpha = 0\) and \(\beta = \frac{\pi}{2}\). The region R can be described as \(0\leq r \leq2\). To find the area A, integrate with respect to θ: $$A = \frac{1}{2} \int_{0}^{\frac{\pi}{2}} \int_{0}^{2}r^2 dr d\theta$$

Evaluate the Integral for the Area

Next, we need to evaluate the double integral for the area A. First, let's integrate with respect to r: $$A = \frac{1}{2} \int_{0}^{\frac{\pi}{2}} [\frac{1}{3}r^3]_{0}^{2} d\theta = \frac{1}{2} \int_{0}^{\frac{\pi}{2}} \frac{8}{3} d\theta$$ Now, integrate with respect to θ: $$A = \frac{1}{2} [\frac{8}{3}\theta]_{0}^{\frac{\pi}{2}} = \frac{4}{3}\pi$$

Set up the Integral for Centroid Coordinates

Now, we need to find the centroid coordinates x and y. We'll need two double integrals for this. The centroid coordinates can be described as follows: $$x = \frac{1}{A} \int_{\alpha}^{\beta} \int_{\rho}^{\varrho} r^2\cos\theta d\theta dr$$ $$y = \frac{1}{A} \int_{\alpha}^{\beta} \int_{\rho}^{\varrho} r^2\sin\theta d\theta dr$$ Where α, β, ρ, and ρ are the same limits established earlier. In this case, \(0\leq\theta\leq\frac{\pi}{2}\) and \( 0\leq r\leq 2\). Set up both integrals: $$x = \frac{1}{A} \int_{0}^{\frac{\pi}{2}} \int_{0}^{2} r^2\cos\theta dr d\theta$$ $$y = \frac{1}{A} \int_{0}^{\frac{\pi}{2}} \int_{0}^{2} r^2\sin\theta dr d\theta$$

Evaluate the Integrals for Centroid Coordinates

Now, we need to evaluate the double integrals for the centroid coordinates x and y. First, let's integrate with respect to r for both integrals: $$x = \frac{1}{A} \int_{0}^{\frac{\pi}{2}} [\frac{1}{3}r^3\cos\theta]_{0}^{2} d\theta = \frac{1}{A} \int_{0}^{\frac{\pi}{2}} \frac{8}{3}\cos\theta d\theta$$ $$y = \frac{1}{A} \int_{0}^{\frac{\pi}{2}} [\frac{1}{3}r^3\sin\theta]_{0}^{2} d\theta = \frac{1}{A} \int_{0}^{\frac{\pi}{2}} \frac{8}{3}\sin\theta d\theta$$ Now, integrate with respect to θ: $$x = \frac{1}{A} [\frac{8}{3} \sin\theta]_{0}^{\frac{\pi}{2}} = \frac{1}{A} \cdot\frac{8}{3}$$ $$y = \frac{1}{A} [-\frac{8}{3} \cos\theta]_{0}^{\frac{\pi}{2}} = \frac{1}{A} \cdot\frac{8}{3}$$

Centroid Coordinates

Lastly, use the area value A that we found earlier to substitute it in x and y coordinates: A = \(\frac{4}{3}\pi\).$$x = \frac{1}{(\frac{4}{3}\pi)} \cdot\frac{8}{3} = \frac{2}{\pi}$$ $$y = \frac{1}{(\frac{4}{3}\pi)} \cdot\frac{8}{3} = \frac{2}{\pi}$$ Thus, the centroid of the given region R is \((\frac{2}{\pi}, \frac{2}{\pi})\).

Key concepts

Centroid

The concept of a centroid is crucial in geometry and physics. It refers to the geometric center of a plane region, often termed as the "center of mass" for an object with uniform density. For symmetrical shapes, such as circles or rectangles, the centroid can be intuitively located by eye. However, complex shapes, like the quarter-circular disk in our exercise, require calculus to find the centroid accurately.

In exercises involving polar coordinates, finding the centroid involves calculating the average position of all points in the plane region. This is done by evaluating the integrals for the centroid's x and y coordinates. These coordinates represent the balance point of the area, ensuring that the shape would evenly balance if supported at this point.

Mathematically, the x and y components of the centroid are determined using the formulas:

• For x-coordinate: \[ x = \frac{1}{A} \int_\alpha^\beta \int_\rho^\varrho r^2 \cos\theta \, dr \, d\theta \]
• For y-coordinate: \[ y = \frac{1}{A} \int_\alpha^\beta \int_\rho^\varrho r^2 \sin\theta \, dr \, d\theta \]

where each element represents a specific integration parameter needed to determine the exact middle point of the shape.

Double Integral

Double integrals play an essential role in calculating areas and volumes, especially within irregularly shaped regions. When analyzing plane regions, such as the quarter-circular disk, double integrals allow us to compute not only the area but also related properties like centroids.

In the context of polar coordinates, a double integral is used to perform two layers of integration: one with respect to the radial coordinate (r) and another with respect to the angular coordinate (θ). This layering helps determine quantities that change over a two-dimensional region.

To find the area A of a polar region, the formula used is:

• \[ A = \frac{1}{2} \int_\alpha^\beta \int_\rho^\varrho r^2 \, dr \, d\theta \]

This involves determining bounds for both r and θ, integrating once with respect to r and then θ, or vice versa, depending on the function. In our example, the quarter-circular disk's limits are 0 to 2 for r and 0 to π/2 for θ, allowing us to compute the exact area comprehensively.

Quarter-Circular Disk

A quarter-circular disk is a sector of a circle that represents one-fourth of the entire circular area.

Such shapes frequently appear in exercises involving polar coordinates due to their simplicity and symmetry, which makes them ideal candidates for understanding basic calculus concepts.

The quarter-circular disk in this problem is defined by the region R in polar coordinates, \( R = \{ (r, \theta): 0 \leq r \leq 2, 0 \leq \theta \leq \frac{\pi}{2} \} \).

Here:

• The radius extends from 0 to 2, representing the length from the origin outward.
• The angle θ spans from 0 to π/2, marking the sector spanning one-fourth of the full circle.

This shape and its allocation in a particular coordinate system allow us to apply integration methods easily to calculate properties like area or centroid.

Plane Region

In polar coordinates, a plane region refers to a specific area of the plane defined by certain boundary conditions of r (radius) and θ (angle).

These regions are frequently used in calculus and mathematical analysis for integrating functions over a specified area.

For example, our exercise involves the plane region of a quarter-circular disk. This is described by the interchanging values of r and θ that succinctly define the moving boundary.

Defining the plane region is crucial as it confines the computation to the specific part of the plane being considered. It avoids unnecessary calculations for portions of the space that are not part of the intended shape. When tackling problems using polar coordinates, identifying the plane region is one of the first steps undertaken, guiding subsequent integration processes and ensuring precise outcomes.

• For simple plane regions, translating between rectangular and polar coordinates can aid in evaluating integrals more efficiently.
• Complex plane regions often require detailed setup in geometrical terms to ensure the analysis aligns with the real-world shape intended.

Understanding the nature of these regions helps simplify calculus operations and derive meaningful solutions for practical applications.