Problem 32 - Thomas Calculus

Question

Integral along different paths Evaluate the line integral \(\int_{C} 2 x \cos y d x-x^{2} \sin y d y\) along the following paths \(C\) in the \(x y\) -plane. a. The parabola \(y=(x-1)^{2}\) from \((1,0)\) to \((0,1)\) b. The line segment from \((-1, \pi)\) to \((1,0)\) c. The \(x\) -axis from \((-1,0)\) to \((1,0)\) d. The astroid \(\mathbf{r}(t)=\left(\cos ^{3} t\right) \mathbf{i}+\left(\sin ^{3} t\right) \mathbf{j}, 0 \leq t \leq 2 \pi,\) counterclockwise from \((1,0)\) back to \((1,0)\)

Step-by-Step Solution

Answer

Paths a, b, and d yield zero; path c gives a value.

Step 1: Parametrization of Path a

For the parabola path given by \( y = (x-1)^2 \), express \( y \) in terms of \( x \) and find the limits for \( x \). The limits for \( x \) are from 1 to 0 (moving from \((1, 0)\) to \((0, 1)\)).

Step 2: Calculating the Line Integral for Path a

Substitute \( y = (x-1)^2 \) into the integral to get \( \int_{1}^{0} \left[ 2x \cos ((x-1)^2) - x^2 \sin ((x-1)^2) \frac{d}{dx}((x-1)^2) \right] \, dx \). Simplify and compute the integral value.

Step 3: Parametrization of Path b

For the line segment path from \((-1, \pi)\) to \((1, 0)\), the line equation is \(y = -\frac{\pi}{2}x + \frac{\pi}{2}\). Parametrize \(x = t\) and \(y = -\frac{\pi}{2}t + \frac{\pi}{2} \) where \( t \) goes from \(-1\) to \(1\).

Step 4: Calculating the Line Integral for Path b

Substitute \(y = -\frac{\pi}{2}x + \frac{\pi}{2}\) into the integral. Compute \(\int_{-1}^{1} \left( 2x \cos\left(-\frac{\pi}{2}x + \frac{\pi}{2}\right) dx - x^2 \sin\left(-\frac{\pi}{2}x + \frac{\pi}{2}\right) \left(-\frac{\pi}{2}\right) dy \right)\). Simplify and calculate the integral.

Step 5: Calculating the Line Integral for Path c

For the path along the \(x\)-axis from \((-1, 0)\) to \((1, 0)\), \(y = 0\), thus the integral simplifies to \(\int_{-1}^{1} 2x dx\). Compute the integral.

Step 6: Parametrization of Path d

For the astroid, use the parametric equations \(x = \cos^3(t)\), \(y = \sin^3(t)\). Differentiate to find \(\frac{dx}{dt} = -3\cos^2(t)\sin(t)\) and \(\frac{dy}{dt} = 3\sin^2(t)\cos(t)\).

Step 7: Calculating the Line Integral for Path d

Substitute \(x = \cos^3(t)\) and \(y = \sin^3(t)\) in the integral and express \(dx\) and \(dy\) in terms of \(dt\). Compute the integral over one complete period (from \(0\) to \(2\pi\)). Due to symmetry, the integral evaluates to zero over the closed loop.

Key Concepts

Parametrization

Parametrization is essential when dealing with line integrals because it allows us to express the curve or path in a mathematical form that simplifies integration. When we parametrize a path, we represent both the x and y coordinates as functions of a new variable, often called the parameter (usually denoted by \(t\)). This makes it easier to perform the integration along the path of interest.

For example, consider the problem of evaluating line integrals along a parabola, a line segment, and an astroid. Each path requires a different approach to parametrization:

• **Parabola:** Here, the path is already defined in terms of \(y\) as a function of \(x\), \(y = (x-1)^2\). The \(x\) coordinate itself acts as the parameter here, as \(x\) naturally ranges from 1 to 0 for the given path.
• **Line Segment:** For a linear path from \((-1, \pi)\) to \((1, 0)\), we use a linear transformation as a parameter where \(x = t\) and \(y = -\frac{\pi}{2}t + \frac{\pi}{2}\). The parameter \(t\) runs from -1 to 1.
• **Astroid:** This curve is nicely represented using trigonometric functions: \(x = \cos^3(t)\) and \(y = \sin^3(t)\), where \(t\) spans from 0 to \(2\pi\). This parametrization exploits the symmetry of the astroid to simplify calculations.
  

Understanding parametrization well allows us to convert complex curves into manageable integrals by finding the right parameter to describe our path.

Path Integration

Path integration is the process of evaluating integrals along a path or curve in the coordinate space. It extends the idea of integrals from one-dimensional intervals to curves, capturing the cumulative effect of a function along these paths. Each path is treated as if it were a collection of infinitesimally small line segments, and we sum the contributions along these segments.

In this exercise, we learn to compute line integrals over several kinds of paths. Key points regarding path integration include:

• **Direction Matters:** The path direction affects the integral's outcome, especially when it involves closed loops or when the function is not constant.
• **Function Components:** The function to be integrated often comprises an expression in terms of both \(x\) and \(y\). For example, the original integral \(\int_{C} 2x\cos y\, dx - x^2\sin y\, dy\) accounts for both coordinate directions.
• **Path Specification:** Understanding the path's nature (whether open or closed) and its specific equation is vital.
  

Path integration thus enables us to measure quantities like work done by a force field or fluid flow along a path, providing a richer interpretation than single-variable integrals.

Calculus

Calculus forms the backbone of line integration, dealing primarily with differentiation and integration. These concepts are extended into multi-dimensions with line integrals and parametrization.

• **Differentiation:** When parametrizing paths like an astroid, we often differentiate the parametric equations to relate changes in \(x\) and \(y\) to changes in \(t\). For instance, differentiating \(x = \cos^3(t)\) gives us \(\frac{dx}{dt} = -3\cos^2(t)\sin(t)\).
• **Integration:** Once parametrization is complete, integration transforms the path into an integral expression with respect to the parameter. This allows us to consider not just direct paths, but complex trajectories like curves and loops.
• **Symmetry and Evaluation:** For path d in the exercise, symmetry helps simplify evaluations. Because the function is well-behaved and the path is a closed loop, it results in zero, showcasing the power of calculus to identify and exploit such properties.
  

Grasping these elements of calculus ensures an effective approach to solving line integrals, making them an essential tool in higher mathematics and applied fields.