Question

The area of a region \(R\) in the plane, whose boundary is the closed curve \(C,\) may be computed using line integrals with the formula $$\text { area of } R=\int_{C} x d y=-\int_{C} y d x$$ These ideas reappear later in the chapter. Let \(R\) be the rectangle with vertices \((0,0),(a, 0),(0, b),\) and \((a, b),\) and let \(C\) be the boundary of \(R\) oriented counterclockwise. Use the formula \(A=\int_{C} x d y\) to verify that the area of the rectangle is \(a b.\)

Short answer

Based on the step-by-step solution above, provide a short answer to the question. Answer: Using the given formula and computing the line integral along the boundary curve of the rectangle, we have found that the area of the rectangle with vertices (0,0), (a, 0), (0, b), and (a, b) is indeed ab.

Step-by-step solution

Setting up the integral for the boundary curve

The boundary curve \(C\) of the rectangle can be split into four line segments: The bottom line segment from \((0,0)\) to \((a,0)\), the right line segment from \((a,0)\) to \((a,b)\), the top line segment from \((a,b)\) to \((0,b)\), and the left line segment from \((0,b)\) to \((0,0)\). We will compute the line integral of \(x\) with respect to \(dy\) along each line segment and sum them up to obtain the integral along the entire boundary curve \(C\).

Evaluate the integral for the bottom line segment

For the bottom line segment from \((0,0)\) to \((a,0)\), the \(y\)-coordinate remains \(0\). Thus, \(dy = 0\), and we have: $$\int_{(0,0)}^{(a,0)} x dy = \int_0^a x\cdot0 dx = 0$$

Evaluate the integral for the right line segment

For the right line segment from \((a,0)\) to \((a,b)\), the \(x\)-coordinate remains \(a\). Thus, we have: $$\int_{(a,0)}^{(a,b)} x dy = \int_0^b a dy = a\int_0^b dy$$ We can evaluate this integral: $$a\int_0^b dy = a[ y]_{0}^{b} = a(b - 0) = ab$$

Evaluate the integral for the top line segment

For the top line segment from \((a,b)\) to \((0,b)\), the \(y\)-coordinate remains \(b\). Thus, \(dy = 0\), and we have: $$\int_{(a,b)}^{(0,b)} x dy = \int_a^0 x\cdot0 dx = 0$$

Evaluate the integral for the left line segment

For the left line segment from \((0,b)\) to \((0,0)\), the \(x\)-coordinate remains \(0\). Thus, we have: $$\int_{(0,b)}^{(0,0)} x dy = \int_b^0 0 dy = \int_b^0 0 dy = 0$$

Add up the integrals

Now we add up the integrals from the four line segments along the boundary curve \(C\): $$A=\int_{C} x dy = \int_{(0,0)}^{(a,0)} x dy + \int_{(a,0)}^{(a,b)} x dy + \int_{(a,b)}^{(0,b)} x dy + \int_{(0,b)}^{(0,0)} x dy$$ Substituting the values we found in Steps 2 to 5, we obtain: $$A=0+ab+0+0=ab$$ Thus, the area of the rectangle with vertices \((0,0),(a, 0),(0, b),\) and \((a, b)\) is indeed \(ab\).

Key concepts

Area Calculation with Line Integrals

The beauty of using line integrals to calculate areas lies in the elegant translation of a geometrical problem into an analytical solution. For a given region \( R \) with a closed curve \( C \) as its boundary, two integral formulations can determine the area of \( R \). These integrals are \( \int_C x \, dy \) and \( -\int_C y \, dx \). By working through such integrals, the calculated result encapsulates the entirety of region \( R \) without directly doubling dimensions or slicing paths.

Consider a rectangle, which is a simple and familiar shape. Its area is typically straightforward to find using base and height dimensions. Yet, employing line integrals introduces a different approach that can be generalized for more complex shapes.

• Integrating directly recognizes the accumulation of change in the desired direction (such as \( dy \) for a region extending orthogonally to the x-axis).
• This method enhances understanding of how areas can be derived using calculus, stepping beyond basic geometry.

Understanding the Closed Curve

A closed curve is a path that begins and ends at the same point, effectively enshrining a space wholly. In the line integral approach to finding an area, the particular form of a closed curve provides a necessary containment of the space.

For a rectangle's boundary \( C \), this path can be segmented into separate but contiguous linear segments that together complete the full envelope of the rectangle. Every line segment is treated as a piece of the whole, each contributing to balancing the integrals:

• The bottom and top horizontal edges affect no area change as they maintain a constant \( y \) coordinate, hence yielding zero energy from their integration.
• The vertical edges (left to right) lead to cumulative area, allowing integration where \( y \) varies.

Thus, understanding closed curves helps in recognizing the cumulative effect of traversing each part, consciously managing these contributions to find complete solutions.

The Importance of Boundary Orientation

Correct orientation of the boundary is crucial, as it ensures that each contribution to the integral acts positively to sum the area. Conventionally, a counterclockwise path is chosen for boundary operations, making integrations conform comfortably to positive mathematics conventions.

Why choose counterclockwise? It aligns with right-hand rules in physics and vector fields, prompting naturally harmonious results when computing vector-aligned integrals.

• The direction affects the positive or negative sign in the resulting line integrals, impacting whether area accumulations are properly calculated.
• A wrong orientation might lead to subtraction where addition is intended, nullifying or reversing essential components.

So, affirming a counterclockwise path ensures all contributions sit correctly, stacking their results to provide an accurate, trustworthy measure of area encased by the curve.