Question

(1 point) Let C be the positively oriented square with vertices (0,0)(0,0), (1,0)(1,0), (1,1)(1,1), (0,1)(0,1). Use Green's Theorem to evaluate the line integral ∫C1y2xdx+9x2ydy∫C1y2xdx+9x2ydy.

Answer 1

Answer:

$\displaystyle \oint_C {y^2x \, dx + 9x^2y \, dy} = \boxed{\bold{4}}$

General Formulas and Concepts:
Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Multiplied Constant]:
$\displaystyle \bold{(cu)' = cu'}$
Derivative Rule [Basic Power Rule]:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Integration

• Integrals

Integration Rule [Fundamental Theorem of Calculus 1]:
$\displaystyle \bold{\int\limits^b_a {f(x)} \, dx = F(b) - F(a)}$

Integration Property [Multiplied Constant]:
$\displaystyle \bold{\int {cf(x)} \, dx = c \int {f(x)} \, dx}$

Multivariable Calculus

Partial Derivatives

Vector Calculus

Circulation Density:
$\displaystyle \bold{F = M \hat{\i} + N \hat{\j} \rightarrow \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y}}$

Green's Theorem [Circulation Curl/Tangential Form]:
$\displaystyle \bold{\oint_C {F \cdot T} \, ds = \oint_C {M \, dx + N \, dy} = \iint_R {\bigg( \frac{\partial N}{\partial x} - \frac{\partial M}{\partial y} \bigg)} \, dx \, dy}$

Step-by-step explanation:

Step 1: Define

Identify given.

$\displaystyle \oint_C {y^2x \, dx + 9x^2y \, dy}$

[See Graph Attachment] Points (0, 0) → (1, 0) → (1, 1) → (0, 1)

↓

$\displaystyle \text{Region:} \left \{ {{0 \leq x \leq 1} \atop {0 \leq y \leq 1}} \right.$

Step 2: Integrate Pt. 1

1. Define vector functions M and N:
   $\displaystyle M = y^2x , \ N = 9x^2y$
2. [Circulation Density] Differentiate [Derivative Rules and Properties]:
   $\displaystyle \frac{\partial M}{\partial y} = 2yx , \ \frac{\partial N}{\partial x} = 18xy$
3. [Green's Theorem] Substitute in Circulation Density:
   $\displaystyle \oint_C {y^2x \, dx + 9x^2y \, dy} = \iint_R {18xy - 2xy} \, dx \, dy$
4. Simplify:
   $\displaystyle \oint_C {y^2x \, dx + 9x^2y \, dy} = \iint_R {16xy} \, dx \, dy$
5. [Integrals] Substitute in region R:
   $\displaystyle \oint_C {y^2x \, dx + 9x^2y \, dy} = \int\limits^1_0 \int\limits^1_0 {16xy} \, dx \, dy$

Step 3: Integrate Pt. 2

We can evaluate the Green's Theorem double integral we found using basic integration techniques listed above:
$\displaystyle \begin{aligned}\oint_C {y^2x \, dx + 9x^2y \, dy} & = \int\limits^1_0 \int\limits^1_0 {16xy} \, dx \, dy \\& = \int\limits^1_0 {8x^2y \bigg| \limits^{x = 1}_{x = 0}} \, dy \\& = \int\limits^1_0 {8y} \, dy \\& = 4y^2 \bigg| \limits^{y = 1}_{y = 0} \\& = \boxed{\bold{4}}\end{aligned}$

∴ we have evaluated the line integral using Green's Theorem.

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Learn more about Green's Theorem: https://brainly.com/question/17186812
Learn more about multivariable calculus: https://brainly.com/question/14502499

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Topic: Multivariable Calculus

Unit: Green's Theorem and Surfaces