Question

In Section 10.3 it was shown that the area of a region enclosed by the polar curve \(r=g(\theta)\) and the rays \(\theta=\alpha\) and \(\theta=\beta,\) where \(\beta-\alpha \leq 2 \pi,\) is \(A=\frac{1}{2} \int_{\alpha}^{\beta} r^{2} d \theta .\) Prove this result using the area formula with double integrals.

Short answer

Question: Prove that the area of a region enclosed by a polar curve $r=g(\theta)$ and the rays $\theta=\alpha$ and $\theta=\beta$ is given by $A=\frac{1}{2}\int_{\alpha}^{\beta} r^{2} d\theta$. Solution: We found that the area is given by the double integral of the Jacobian $g(\theta)$ and the transformation from polar to rectangular coordinates. After evaluation, we obtained the formula $A=\frac{1}{2}\int_{\alpha}^{\beta} r^2 d\theta$. Therefore, the area of the region enclosed by the polar curve and rays is equal to $\frac{1}{2}\int_{\alpha}^{\beta} r^2 d\theta$.

Step-by-step solution

Convert the polar curve to rectangular coordinates

First, we need to convert the polar curve \(r=g(\theta)\) into rectangular coordinates using the equations \(x=r\cos(\theta)\) and \(y=r\sin(\theta)\). Substitute \(r=g(\theta)\) in both these equations: \(x = g(\theta)\cos(\theta)\) \(y = g(\theta)\sin(\theta)\)

Find the Jacobian

The next step is to find the Jacobian of the transformation from polar to rectangular coordinates. The Jacobian is given by: $J=\frac{\partial(x,y)}{\partial(r,\theta)}=\begin{vmatrix}\frac{dx}{dr} & \frac{dx}{d\theta}\\ \frac{dy}{dr} & \frac{dy}{d\theta}\end{vmatrix}$ Taking partial derivatives, we get: \(\frac{dx}{dr} = \cos(\theta)\), \(\frac{dx}{d\theta} = -g(\theta)\sin(\theta) + g'(\theta)\cos(\theta)\) \(\frac{dy}{dr} = \sin(\theta)\), \(\frac{dy}{d\theta} = g(\theta)\cos(\theta) + g'(\theta)\sin(\theta)\) Now, find the determinant: $J=\begin{vmatrix}\cos(\theta) & -g(\theta)\sin(\theta) + g'(\theta)\cos(\theta)\\ \sin(\theta) & g(\theta)\cos(\theta) + g'(\theta)\sin(\theta)\end{vmatrix}$ \(J=\cos(\theta)(g(\theta)\cos(\theta) + g'(\theta)\sin(\theta)) - \sin(\theta)(-g(\theta)\sin(\theta) + g'(\theta)\cos(\theta))\) Simplify and notice that the derivatives of \(g(\theta)\) cancel out: \(J = g(\theta)(\cos^2(\theta) + \sin^2(\theta)) = g(\theta)\)

Setup the double integral

Now that we have the Jacobian, we can set up the double integral in rectangular coordinates using the area formula. Recall that an infinitesimal area in polar coordinates is given by \(dA = r\:dr\:d\theta\). Since \(r = g(\theta)\), we have \(dA = g(\theta)\:dr\:d\theta\), and since the Jacobian \(J = g(\theta)\): \(dA = J\:dr\:d\theta = g(\theta)\:dr\:d\theta\) The area enclosed by the polar curve \(r = g(\theta)\) and rays \(\theta = \alpha\) and \(\theta = \beta\) is given by the double integral of \(dA\): \(A = \int_{\alpha}^{\beta}\int_{0}^{g(\theta)} g(\theta)\:dr\:d\theta\)

Evaluate the double integral

We can evaluate the integral with respect to \(r\) first: \(A = \int_{\alpha}^{\beta}\frac{1}{2}(g(\theta))^2 d\theta\) Here, we see that our result is the same as given in the exercise: \(A = \frac{1}{2}\int_{\alpha}^{\beta}r^{2} d\theta\) Thus, we have proved that the area of a region enclosed by the polar curve \(r=g(\theta)\) and the rays \(\theta=\alpha\) and \(\theta=\beta\) is given by the formula \(A=\frac{1}{2} \int_{\alpha}^{\beta} r^{2} d \theta\).

Key concepts

Double Integrals

Double integrals extend the concept of a single integral to two dimensions, allowing us to calculate the volume under a surface or the area of a region in the xy-plane. In the context of polar coordinates, double integrals consider both the radial distance, denoted by the variable 'r', and the angle, denoted by '\(\theta\)', to evaluate the area enclosed by a curve. To compute an area using double integrals, we set up an integral that sums the infinitesimal area elements '\(dA\)' over the region of interest.

When we transition from Cartesian coordinates to polar coordinates, these infinitesimal elements change from rectangular '\(dxdy\)' slices to sectors of a circle with area '\(r dr d\theta\)'. This reflects the idea that, at a given radius, moving a small distance 'dr' along the radial direction and a small angle 'd\theta' along the arc results in a small area that is part of a disk, the size of which depends on the radius 'r' and the infinitesimal changes 'dr' and 'd\theta'.

Jacobian Determinant

The Jacobian determinant is an essential part of transforming double integrals between coordinate systems. It represents how much an area element in one coordinate system, such as polar coordinates, 'stretches' or 'contracts' when mapped to another system, like Cartesian coordinates. Mathematically, the Jacobian is the determinant of the matrix composed of the partial derivatives of the new coordinates with respect to the old ones.

For a transformation from polar to rectangular coordinates, the Jacobian shows how an area element '\(r dr d\theta\)' in polar coordinates translates into an equivalent area element in Cartesian coordinates ('\(dx dy\)'). The absolute value of this determinant is used to adjust the size of the original area element to maintain the equivalence of area across coordinate systems. This is why we multiply by the Jacobian when converting the polar area element to rectangular form '\(J dr d\theta = dx dy\)', ensuring the area calculation is invariant under the change of coordinates.

Polar to Rectangular Coordinate Transformation

Transforming from polar to rectangular coordinates involves converting the polar position defined by a radius 'r' and an angle '\(\theta\)', to its Cartesian counterpart defined by 'x' and 'y'. The transformation uses the relationships '\(x = r\cos(\theta)\)' and '\(y = r\sin(\theta)\)' which are derived from the unit circle and trigonometry.

In the area calculation context, we apply these transformations to the integral limits and the integrand's function. This is essential because while polar coordinates can simplify the integration process for circular or sector-shaped regions, some calculations involving more complex shapes or known functions might be easier to handle in the rectangular system, especially when dealing with partial derivatives, as required for finding the Jacobian determinant.

Area of Polar Regions

The area of polar regions can be found using the concept of double integrals in polar coordinates. Given a polar function '\(r = g(\theta)\)', we define the enclosed area by considering the curve and two rays that emanate from the origin at angles '\(\theta = \alpha\)' and '\(\theta = \beta\)'.

The double integral for the area is set up to sum up all the infinitesimal areas '\(dA = r dr d\theta\)' over the region. The integration proceeds over 'r' from 0 to '\(g(\theta)\)' and over '\(\theta\)' from '\(\alpha\)' to '\(\beta\)'. This captures the sweeping motion of the radius 'r' as it traces the area against the angle '\(\theta\)', effectively filling in the sector or the entire region enclosed by the curve. The result gives a precise measure of the area in polar regions, relevant in fields ranging from physics and engineering to mathematics and graphic design.