Problem 12 - Calculus: Early Transcendental Functions

Question

Use Green's Theorem to evaluate the indicated line integral. \(\oint_{C} \mathbf{F} \cdot d \mathbf{r},\) where \(\mathbf{F}=\left\langle y^{2}+3 x^{2} y, x y+x^{3}\right\rangle\) and \(C\) is formed by \(y=x^{2}\) and \(y=2 x\)

Step-by-Step Solution

Answer

To evaluate this integral using Green's Theorem, calculate the curl of F, identify the region D, and set up and calculate the double integral of curl F over D.

Step 1: Express the vector field

The given vector field is \(\mathbf{F}=\left\langle y^{2}+3 x^{2} y, x y+x^{3}\right\rangle.\)

Step 2: Calculate the curl of F

The curl of F in two dimensions is determined by \(\nabla \times \mathbf{F} = \frac{∂ F_2}{∂x} - \frac{∂ F_1}{∂y}\). Here \(F_1 = y^{2}+3 x^{2} y\) and \(F_2 = x y+x^{3}\). So \(\nabla \times \mathbf{F} = \frac{∂ (x y+x^{3}) }{∂x} - \frac{∂ (y^{2}+3 x^{2} y)}{∂y} = 2x^2 - (2y+3x^2) = 2x^2 - 2y - 3x^2 = -2y -x^2.\)

Step 3: Identify the region D

The curves \(y=x^2\) and \(y=2x\) bound the region D in the xy-plane. If we sketch these curves, we see the region is bounded by \(x^2 ≤ y ≤ 2x\). For \(x^2 = 2x\), we find the limits of x are 0 and 2.

Step 4: Set up and calculate the double integral

By Green's Theorem, the integral of curl F over D is equal to the line integral of F over C: \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}= \int \int_D \nabla \times \mathbf{F} \,dA\). Substituting curl of \(\mathbf{F}\) we get: \(\int_{0}^{2} \int_{x^{2}}^{2x} (-2y -x^2) \, dy \, dx.\) Evaluating this double integral gives the final solution.

Key Concepts

Vector Field

In mathematics and physics, a vector field is a construction that assigns a vector to every point in a space. Imagine a flowing stream where every point has a direction and a magnitude, like arrows showing the direction of flow and their intensity.

In our exercise, the vector field is given as \(\mathbf{F}=\langle y^{2}+3 x^{2} y, x y+x^{3}\rangle\). Here, we have two components:

• \(y^{2}+3 x^{2} y\) influencing direction along the x-axis
• \(x y+x^{3}\) directing along the y-axis

This vector field represents how the values of x and y change across different points in the region. Understanding vector fields is crucial as they provide the basis for more advanced concepts like line integrals and Green's Theorem.

Line Integral

A line integral is a way to integrate functions along a curve. Think of it as summing up values of some vector field components along a path.

For instance, in the given exercise, the line integral \(\oint_{C} \mathbf{F} \cdot d \mathbf{r}\) is evaluated along the curve \(C\), which is formed by \(y=x^{2}\) and \(y=2x\).
The line integral considers both the path and the function applied over it:

• \(\mathbf{F}\): The vector field function
• \(d \mathbf{r}\): A small segment of the curve

Green's Theorem provides a technique to convert this into a double integral over a region, simplifying calculation.

Double Integral

Double integrals extend the concept of integrating a function over a domain in two-dimensional space. It effectively allows us to accumulate values over an area, considering both x and y coordinates.

In Green's Theorem application, instead of calculating a line integral over a path, we calculate a double integral over a region using the curl of a vector field. The bounded region \(D\) in our problem is enclosed by curves \(y=x^2\) and \(y=2x\). We determine the limits of integration within this region by:

• Knowing the outer curve boundary
• Identifying the intersection points of the curves

For our specific problem, the integral is set up as \(\int_{0}^{2} \int_{x^{2}}^{2x} (-2y -x^2) \, dy \, dx\), effectively transforming the line integral into a solvable double integral.

Curl of a Vector Field

The concept of the curl of a vector field encompasses understanding how a vector field rotates. In physics, it can be viewed as the swirling strength of the field at a certain point.

Green's Theorem, used in our exercise, involves calculating this curl to convert a line integral into a double integral. The curl in two dimensions is given by \(abla \times \mathbf{F} = \frac{∂ F_2}{∂x} - \frac{∂ F_1}{∂y}\).
From the solution, the curl \(abla \times \mathbf{F}\) simplifies to \(-2y -x^2\). Here's why that matters:

• Describes the tendency to rotate within a small region of the vector field
• Directly relates to how easily the path integral can be transformed using Green's Theorem

Understanding curl is fundamental when dealing with vector fields, as it characterizes how the field behaves beyond just magnitude or direction.