Question

Write the double integral \(\iint_{R} f(x, y) d A\) as an iterated integral in polar coordinates when \(R=\\{(r, \theta): a \leq r \leq b, \alpha \leq \theta \leq \beta\\}.\)

Short answer

Question: Write the double integral of the function f(x, y) over the region R in polar coordinates, given the boundaries a ≤ r ≤ b and α ≤ θ ≤ β. Answer: The double integral of f(x, y) over the region R in polar coordinates is: $$\iint_R f(x, y) dA = \int_\alpha^\beta \int_a^b f(r\cos(\theta), r\sin(\theta))r\,dr d\theta$$.

Step-by-step solution

Identify the given region and function

In the exercise, the region R is given in terms of polar coordinates with bounds a ≤ r ≤ b and α ≤ θ ≤ β. We are asked to find the double integral of the function f(x, y) over this region.

Write the double integral with respect to polar coordinates

Since we are converting to polar coordinates, we will use r and θ as variables for integration. The double integral can be written as: $$\iint_R f(x, y) dA = \iint_R f(r\cos(\theta), r\sin(\theta))r\,dr d\theta$$ Here, we have multiplied by the Jacobian, which is r in the case of polar coordinates.

Set the limits of integration for r and θ

Given that a ≤ r ≤ b and α ≤ θ ≤ β, the limits for the iterated integral are straightforward. The limits for r will be from a to b, and the limits for θ will be from α to β. This gives us the following iterated integral: $$\iint_R f(x, y) dA = \int_\alpha^\beta \int_a^b f(r\cos(\theta), r\sin(\theta))r\,dr d\theta$$

Evaluate the iterated integral

To find the value of the given double integral, you would need to evaluate the iterated integral for the specific function f(x, y). However, without knowing the specific function, the exercise only requires us to set up the iterated integral using the polar coordinates for the given region R. The final result is: $$\iint_R f(x, y) dA = \int_\alpha^\beta \int_a^b f(r\cos(\theta), r\sin(\theta))r\,dr d\theta$$

Key concepts

Polar Coordinates

Polar coordinates are an alternative coordinate system used primarily in the fields of mathematics and physics. Instead of determining a point's position with traditional x and y coordinates, we use the distance from a reference point and an angle from a reference direction. In this coordinate system, any point on a plane can be located using:

• Radius (r): The distance from the origin (0, 0) to the point.
• Angle (θ): The angle measured counterclockwise from the positive x-axis.

This system is particularly useful when dealing with problems that have radial symmetry, like circular regions or when converting complex Cartesian interactions to simpler polar interactions. For any point in polar coordinates, its Cartesian counterparts can be given as:

• \(x = r \cos(\theta)\)
• \(y = r \sin(\theta)\)

Understanding how to switch between these two systems is vital when applying polar coordinates in double integrals.

Iterated Integral

An iterated integral refers to evaluating multiple integrals in a sequence, typically with respect to several variables. In double integrals, iterated integrals allow us to break down the evaluation process into smaller, more manageable steps by integrating over each variable one at a time.

• Start by integrating the innermost function with respect to one variable, like \(r\).
• Subsequently, integrate the result with respect to the other variable, like \(\theta\).

When using polar coordinates, the iterated integral format becomes crucial for setting up and solving integrals over regions defined in terms of \(r\) and \(θ\). By using the iterated integral form, we can structure the computation as:\[\int_{\alpha}^{\beta} \int_{a}^{b} f(r \cos(\theta), r \sin(\theta)) r\,dr d\theta\]This approach helps to simplify many integration problems, particularly those involving circular or rotationally symmetric regions.

Jacobian in Polar Coordinates

The Jacobian is a mathematical factor that accounts for the stretching or compression of areas when changing from one coordinate system to another, such as from Cartesian to polar coordinates. In two dimensions, the Jacobian is a determinant of the matrix comprising partial derivatives. It measures how a small area transforms under the change of variables.

• For polar coordinates, the Jacobian is given by the expression: \(r\).
• It appears when setting up the integral to account for how much each infinitesimal area element \(dA\) in Cartesian coordinates is affected in polar coordinates.

In polar coordinates, the differential area element \(dA\) transforms to \(r\, dr d\theta\), where \(r\) is the Jacobian. This adjustment is essential for correctly evaluating integrals over polar regions.

Limits of Integration

The limits of integration define the range over which we evaluate an integral. In polar coordinates, these limits are typically given in terms of \(r\) and \(θ\), reflecting the nature of the region of interest.
When you understand the geometric region you are working with, you can set the appropriate limits for each variable:

• For the variable \(r\), determine the minimum and maximum radius values that capture the region. Here, \(a \leq r \leq b\).
• For the angle \(θ\), identify the angular bounds, typically set from \(α\) to \(β\), covering the desired rotational sweep.

When these limits are plugged into the iterated integral, they specify exactly which part of the coordinate plane is being integrated over. This helps ensure that the computation respects the predefined region, allowing you to apply the integral accurately to capture the area or volume.