Question

Area line integral In terms of the parameters \(a\) and \(b\), how is the value of \(\oint_{C}\) ay \(d x+\) bx \(d y\) related to the area of the region enclosed by \(C\), assuming counterclockwise orientation of \(C ?\)

Short answer

Based on the given integral expression, we can conclude that the value of the integral \(\oint_{C} (ay \, dx + bx \, dy)\) is related to the area of the region enclosed by C through the factor \((b-a)\). This relationship is derived using Green's theorem and evaluating the integral over the area D.

Step-by-step solution

Recall the definition of a counterclockwise area line integral and identify the given integral's form

The area enclosed by a curve C, with a counterclockwise orientation, can be computed using Green's theorem. Given an integral of the form: \[ \oint_{C} (P dx + Q dy) = \iint_{D} (\frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}) dA \] where P and Q are two functions with continuous partial derivatives, and D is the area enclosed by the curve C. In our case, the given integral is: \[ \oint_{C} (ay \, dx + bx \, dy) \] Now, let's identify the necessary partial derivatives.

Compute the partial derivatives

Compute the partial derivatives as follows: \[ \frac{\partial Q}{\partial x} = \frac{\partial (bx)}{\partial x} = b \] and \[ \frac{\partial P}{\partial y} = \frac{\partial (ay)}{\partial y} = a \] Now, we can use Green's theorem to compute the area enclosed by C.

Apply Green's theorem to compute the area enclosed by C

Using Green's theorem, we can rewrite the given integral as: \[ \oint_{C} (ay \, dx + bx \, dy) = \iint_{D} (b - a) dA \] Now, we want to find the relationship between the value of the integral and the area of the region enclosed by C. To do this, we can integrate over the area D: \[ \iint_{D} (b-a) dA = (b-a) \iint_{D} dA \]

Compare the calculated value to find the relationship between the two

Since the integral value is equal to \((b-a)\) times the area of the region enclosed by C, we can conclude that the value of the integral \(\oint_{C} (ay \, dx + bx \, dy)\) is related to the area of the region enclosed by C through the factor \((b-a)\).

Key concepts

Line Integral

A line integral is an integral where the integration is performed over a curve. It involves calculating a sum along the path or curve in a vector field. In mathematical terms, you often see line integrals expressed as \( \oint_{C} (P \, dx + Q \, dy) \). Here, \( P \) and \( Q \) are functions of two-dimensional space, and \( dx \) and \( dy \) represent infinitesimal changes along the x and y directions, respectively.

In practical terms, the line integral can be thought of as the total accumulation of certain quantities along the curve, such as work done by a force or flux across a boundary.

• The curve \( C \) is typically parametrized, meaning its path can be expressed using a parameter like \( t \).
• The integral \( \oint \) denotes an integral over a closed curve like a circle or loop.

Understanding line integrals is crucial because they are used in various fields, from physics to engineering, allowing the calculation of quantities related to paths in vector fields.

Area Enclosed by Curve

The area enclosed by a curve is one key application of Green's Theorem, which connects a line integral around a closed curve with a double integral over the area it encompasses.

In the context of this problem, the enclosed area of a curve \( C \) with a specific line integral can be deduced using the relation provided by Green's Theorem:

• The theorem equates the value of the line integral around the region's boundary with a double integral over the region itself.
• In terms of the given problem, applying Green's Theorem transforms the line integral into a form focused on calculating the enclosed area.

Thus, the line integral equation \( \oint_{C} (ay \, dx + bx \, dy) \) relates to the area through its transformation into a double integral of consistent functions, translating to \( (b-a) \cdot \text{Area} \). This relationship helps in determining the physical space within the boundary, offering a valuable analysis tool in geometry and physics.

Partial Derivatives

Partial derivatives are derivatives taken with respect to one variable while keeping others constant, a fundamental concept in multivariable calculus. When dealing with a function of several variables, like \( f(x, y) \), you can compute derivatives with respect to each variable individually.

In the case of Green's Theorem, partial derivatives help in evaluating the expressions \( \frac{\partial Q}{\partial x} \) and \( \frac{\partial P}{\partial y} \):

• \( \frac{\partial Q}{\partial x} \) represents the derivative of function \( Q \) with respect to \( x \), treating \( y \) as constant.
• \( \frac{\partial P}{\partial y} \) implies a derivative of \( P \) with respect to \( y \), holding \( x \) fixed.

These values are plugged into Green's Theorem to connect the line integral with the double integral over area \( D \). It's crucial to manage these derivatives correctly to apply Green's Theorem and solve for areas or other properties linked to the boundary of a region.