Question

Verify Green's Theorem by evaluating both integrals $$ \int_{C} y^{2} d x+x^{2} d y=\int_{R} \int\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right) d A \text { for the given path. } $$ C: \text { square with vertices }(0,0),(4,0),(4,4),(0,4) $$

Short answer

The results from the line integrals and the double integral are equal, therefore verifying Green's Theorem.

Step-by-step solution

Define M and N

From the integral, identify \( M = y^2 \) and \( N = x^2 \) as the equations you will be working with.

Set up the Line Integrals around C

Parametrize the integers over the path C as a piecewise-defined function. There are four segments in this case: from (0,0) to (4,0), the line is \( x = t, y = 0 \) for \( t \in [0,4] \); from (4,0) to (4,4), the line is \( x = 4, y = t \) for \( t \in [0,4] \); from (4,4) to (0,4), the line is \( x = t, y = 4 \) for \( t \in [4,0] \); from (0,4) to (0,0), the line is \( x = 0, y = t \) for \( t \in [4,0] \).

Calculation of the Line Integrals

Calculate the line integrals for each piece using the parametrizations from Step 2. Then sum the results of the line integrals.

Calculation of the Double Integral

Calculate the partial derivatives of \( M \) and \( N \) with respect to \( y \) and \( x \), respectively. Subtract them and set up the double integral of the resulting function over the region \( R \). Here \( R \) is the square defined by \( 0 \leq x \leq 4, 0 \leq y \leq 4 \)

Evaluate the Double Integral

Evaluate the integral obtained in Step 4 by applying the basic rules of integration.

Verify Green’s Theorem

Compare the results from the line integrals (Step 3) and the double integral (Step 5). If they are equal, it verifies Green’s theorem.