Problem 39 - Thomas Calculus

Question

Area as a line integral Show that if \(R\) is a region in the plane bounded by a piecewise smooth, simple closed curve \(C,\) then $$\begin{array}{l}{\text {Area of}}\end{array}R=\oint_{C} x d y=-\oint_{C} y d x$$

Step-by-Step Solution

Answer

Using Green's Theorem, both line integrals express the area of \( R \).

Step 1: Understand the Theorem

The theorem states the area of a region \( R \), bounded by a piecewise smooth, simple closed curve \( C \), can be calculated using line integrals as \( \oint_{C} x \, dy = -\oint_{C} y \, dx \). This involves parametrizing the boundary \( C \) and setting up appropriate integrals.

Step 2: Parametrize the Curve C

Suppose the curve \( C \) is parametrized by a vector function \( \mathbf{r}(t) = (x(t), y(t)) \) for \( t \) varying over an interval \([a, b]\). The direction of traversal should be counter-clockwise for positive orientation.

Step 3: Apply Green's Theorem

According to Green's Theorem, for a simply connected region \( R \), \( \oint_{C} x \, dy + y \, dx = \int\int_{R} (1 - 1) \, dA = 0 \). Thus, the curve integral simplifies to \( \oint_{C} x \, dy = - \oint_{C} y \, dx \), proving the desired result.

Step 4: Verify Area Integral

Green's Theorem also provides that \( \oint_{C} x \, dy = \int\int_{R} 1 \, dA \), which is the area of \( R \). This verifies that both line integrals indeed describe the area of the region.

Key Concepts

Green's Theorem

Green's Theorem is an essential tool in vector calculus that connects a line integral around a simple closed curve with a double integral over the plane region it encloses. Essentially, Green's Theorem allows us to relate the circulation around the boundary of a region to the sum of the curl over the area it encloses. Specifically, it shows that:

• For a vector field \( \mathbf{F} = (P(x, y), Q(x, y)) \), the line integral \( \oint_C P \, dx + Q \, dy \) is related to the double integral over the region \( R \) by:
  \[ \oint_{C} P \, dx + Q \, dy = \int \int_{R} \left( \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} \right) dA \]

In the context of area calculation, Green's Theorem simplifies the computation of the area of a region \( R \) by using line integrals over its boundary. In particular, selecting \( P = 0 \) and \( Q = x \) reveals the area as \( \oint_{C} x \, dy = \int\int_{R} 1 \, dA \), confirming the area can be calculated by these integrals.

Curve Parametrization

To evaluate the line integrals involved in these calculations, it is necessary to parametrize the curve \( C \). Parametrization involves expressing the curve as a function of a parameter, typically \( t \), over an interval. We typically use a vector function \( \mathbf{r}(t) = (x(t), y(t)) \).
This makes it possible to evaluate integrals by substituting for \( x \) and \( y \) and integrating with respect to \( t \). The interval over which \( t \) varies should cover one complete traversal of the curve. Remember that the orientation matters; for the purposes of Green's Theorem, the path should be traversed counter-clockwise for a positive orientation.

• Example of Parametrization: For a circle of radius \( a \), \( \mathbf{r}(t) = (a\cos(t), a\sin(t)) \) with \( t \) in \([0, 2\pi]\).
• For a rectangle, you would have separate parametrizations for each of the four curve segments.

Area Calculation

Calculating the area of a region within a plane using line integrals is a powerful technique derived from Green's Theorem. When the region \( R \) is bounded by a curve \( C \), you can express the area using line integrals as follows:

• \( \oint_{C} x \, dy = \text{Area of } R \)
• Alternatively, \( -\oint_{C} y \, dx \) gives the same result.

This method leverages the parametrization of the boundary curve and evaluates integrals to obtain the enclosed area.
This approach is highly valuable because it transforms an area calculation into a boundary problem, typically easier to solve by dealing with parametric equations rather than a double integral over the area itself. In practice, you set up and evaluate these line integrals to find the region's area.

Piecewise Smooth Curves

A piecewise smooth curve basically is a curve made up of a finite number of smooth sections joined together.
They are important in calculus and physical applications because they allow you to model real-world scenarios where transitions may not be entirely smooth.

• Each segment of a piecewise smooth curve can be parametrized separately.
• The smoothness of each segment ensures that the line integrals are well-defined.

While the entire curve may have kinks or corners, Green's Theorem can still be applied as the theorem only requires the curve to be piecewise smooth. This property is critical when calculating the area using line integrals, as it allows diverse shapes and practical usage.
For instance, if you have a polygon or a combination of arc and straight lines bounding a region, piecewise smoothness enables appropriate calculations along each individual segment.