Question

use green's theorem to evaluate the line integral along the given positively oriented curve. c x2y2 dx y tan−1(8y) dy, c is the triangle with vertices (0, 0), (1, 0), and (1, 4)

Answer 1

The line integral along the curve C is evaluated using Green's Theorem by converting it to a double integral over the region D bounded by C. The integral evaluates to the double integral of ∂Q/∂x - ∂P/∂y over D, and after calculating, we obtain the result for the line integral.

Step-by-step explanation:

Using Green's Theorem to Evaluate a Line Integral

To solve the line integral question using Green's Theorem, we have the integral of a vector field over the curve C, which is the triangle with vertices (0, 0), (1, 0), and (1, 4). Green's Theorem relates a line integral around a simple closed curve C to a double integral over the plane region D bounded by C.

Firstly, we express the given line integral in the form ∫ C P dx + Q dy, where P = x2y2 and Q = y tan∑(8y). Applying Green's Theorem, we convert this line integral into a double integral over the domain D bounded by C as follows:

∫ C P dx + Q dy = ∬ D (∂Q/∂x - ∂P/∂y) dA.

We find the partial derivatives ∂Q/∂x and ∂P/∂y, evaluate the double integral over D, and hence, get the value of the line integral using the theorem. Since the region D is a triangle, we set up the limits of integration for x from 0 to 1 and for y as a function of x, ranging from 0 to 4x.

In this specific case, we need to calculate the following double integral:

∬ D (0 - 2x2y) dA,

which simplifies the integral to:

∬01 ∬04x (-2x2y) dy dx.

After integrating with respect to y and then x, we obtain the result for the original line integral.

Answer 2

The evaluation of the double integral over the triangular region R results in -3.5.

To evaluate the given line integral over the triangular region R:

Given line integral:

`\[ \oint_C x^2y^2 \, dx + y \tan^(-1) y \, dy \]`

We use Green's Theorem, which relates a line integral over a closed curve to a double integral over the region enclosed by the curve. The theorem is given by:

`\[ \oint_C P \, dx + Q \, dy = \iint_R \left((\partial Q)/(\partial x) - (\partial P)/(\partial y)\right) \, dA \]`

Where P and Q are components of a vector field.

For the given integral:

`\[ P = x^2y^2 \]`

`\[ Q = y \tan^(-1) y \]`

Calculating the partial derivatives:

`\[ (\partial P)/(\partial y) = 2xy^2 \]`

`\[ (\partial Q)/(\partial x) = 0 \]`

Applying Green's Theorem:

`\[ \oint_C x^2y^2 \, dx + y \tan^(-1) y \, dy = \iint_R (0 - 2xy^2) \, dA \]`

The limits of integration for the triangular region R are x from 0 to 1 and y from 0 to 3 - x.

Evaluating the double integral:

`\[ \iint_R -2xy^2 \, dA = \int_0^1 \int_0^(3 - x) -2xy^2 \, dy \, dx \]`

Integrating with respect to y first:

`\[ \int_0^(3 - x) -2xy^2 \, dy = -18x + 18x^2 - 2x^3 \]`

Integrating the result with respect to x from 0 to 1:

`\[ \int_0^1 (-18x + 18x^2 - 2x^3) \, dx \]`

Now, integrate each term separately:

`\[ \int_0^1 -18x \, dx = -18 \int_0^1 x \, dx \]`

`\[ = -18 \left[(x^2)/(2)\right]_0^1 \]`

`\[ = -18 \cdot \left((1^2)/(2) - (0^2)/(2)\right) \]`

`\[ = -18 \cdot \left((1)/(2) - 0\right) \]`

`\[ = -18 \cdot (1)/(2) \]`

`\[ = -9 \]`

`\[ \int_0^1 18x^2 \, dx = 18 \int_0^1 x^2 \, dx \]`

`\[ = 18 \cdot \left[(x^3)/(3)\right]_0^1 \]`

`\[ = 18 \cdot \left((1^3)/(3) - (0^3)/(3)\right) \]`

`\[ = 18 \cdot \left((1)/(3) - 0\right) \]`

`\[ = 18 \cdot (1)/(3) \]`

`\[ = 6 \]`

`\[ \int_0^1 -2x^3 \, dx = -2 \int_0^1 x^3 \, dx \]`

`\[ = -2 \cdot \left[(x^4)/(4)\right]_0^1 \]`

`\[ = -2 \cdot \left((1^4)/(4) - (0^4)/(4)\right) \]`

`\[ = -2 \cdot \left((1)/(4) - 0\right) \]`

`\[ = -2 \cdot (1)/(4) \]`

`\[ = -(1)/(2) \]`

Now, sum up these individual integrals:

`\[ \int_0^1 (-18x + 18x^2 - 2x^3) \, dx = -9 + 6 - (1)/(2) \]`

`= -3.5`

Therefore, the value of the integral
`\( \int_0^1 (-18x + 18x^2 - 2x^3) \, dx \)` is -3.5.

Question:

Use Green's Theorem to evaluate the line integral along the given positively oriented curve.

`\[\oint_(C) x^(2) y^(2) d x+y \tan ^(-1) y d y\]`

where C is the triangle with vertices (0,0),(1,0) and (1,3).