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

The integral of a function \(f(x, y)\) over a region \(R\) defined by polar coordinates \((r, \theta)\) can be expressed as \(\iint_{R} f(r\cos(\theta), r\sin(\theta))r d A\) with limits of integration \(a \leq r \leq b\) and \(\alpha \leq \theta \leq \beta\). The final expression is: $$ \int_{\alpha}^{\beta}\int_{a}^{b} f(r\cos(\theta), r\sin(\theta))r dr d\theta $$

Step-by-step solution

Express \(x\) and \(y\) values in polar coordinates

We need to express the given function \(f(x, y)\) in polar coordinates. We know that \(x = r\cos(\theta)\) and \(y=r\sin(\theta)\). Hence we can rewrite the function \(f(x, y)\) as \(f(r\cos(\theta), r\sin(\theta))\).

Write the integral in polar coordinates

To set up a double integral in polar coordinates, we need to multiply the integrand by the Jacobian, which is equal to \(r\). Therefore, the integral will take the form \(\iint_{R} f(r\cos(\theta), r\sin(\theta))r d A\).

Determine the limits of integration

The region \(R\) is given as \(\{(r, \theta): a \leq r \leq b, \alpha \leq \theta \leq \beta\}\). This means that \(r\) ranges from \(a\) to \(b\) and \(\theta\) ranges from \(\alpha\) to \(\beta\). These will be our limits of integration for the iterated integral.

Set up the iterated integral

We now have all the information needed to set up the iterated integral in polar coordinates. The integral will have limits \(a \leq r \leq b\) and \(\alpha \leq \theta \leq \beta\), and the integrand will be \(f(r\cos(\theta), r\sin(\theta))r\). Therefore, the iterated integral is: $$ \int_{\alpha}^{\beta}\int_{a}^{b} f(r\cos(\theta), r\sin(\theta))r dr d\theta $$

Key concepts

Iterated Integral

An iterated integral is a way to calculate the integral over a multi-dimensional area by breaking it down into successive one-dimensional integrals. In a two-dimensional scenario, this means we integrate one variable while treating the other as a constant, and then integrate the resulting expression with respect to the second variable.

When dealing with double integrals, the iterated integral allows us to compute the area under a surface. By doing each integral consecutively, we can simplify what would otherwise be a complex calculation. This step-by-step integration process can be used in different coordinate systems, including Cartesian and polar coordinates, among others.

Polar Coordinate System

The polar coordinate system is an alternative to the Cartesian coordinate system for representing points in a plane. While Cartesian coordinates use an ordered pair \(x, y)\) to designate a point's horizontal and vertical displacement, polar coordinates use \(r, \theta)\) where \(r\) is the radial distance from the origin (or how far away the point is from the center), and \(\theta\) is the angular coordinate, representing the angle made with the positive x-axis.

Polar coordinates are particularly useful when dealing with problems where symmetry or circular shapes are involved, as they simplify expressions and calculations. For instance, circles around the origin are represented as a constant radial distance, \(r\), which greatly simplifies the integral's limits when in polar form.

Jacobian Determinant

The Jacobian determinant appears in the change of variables process during integration and is a critical concept in multivariable calculus. As functions are transformed from Cartesian to polar coordinates, the area element changes, requiring an adjustment expressed by the Jacobian determinant.

In the polar coordinate system, the Jacobian determinant is \(r\), correlating to the fact that areas in polar coordinates grow with the distance from the origin. In practice, this means when converting a double integral from Cartesian to polar coordinates, we multiply the integrand by \(r\) to account for the 'stretching' of the area element as we move further from the origin. Hence, it ensures that the area transformation is properly scaled in the new coordinate system.

Limits of Integration

The limits of integration define the range over which an integral is evaluated, specifying the start and end points for each variable. When converting to polar coordinates, these limits correspond to the ranges of \(r\) and \(\theta\), the radial and angular components, respectively. The region, \(R\), is typically described by inequalities that determine the minimum and maximum values for \(r\) and \(\theta\).

For example, the region in our exercise is bounded by \(a \leq r \leq b\) and \(\alpha \leq \theta \leq \beta\), which imply that the radial distance \(r\) is confined between two values, and the angle \(\theta\) is confined between two angular directions. Carefully determining these limits is essential, as they directly influence the result of the integral. When setting up the iterated integral in polar coordinates, one must carefully select these limits to properly describe the region over which the function is integrated.