For each given path, verify Greens Theorem by showingthatFor each path, which integral

Chapter 15, Problem 42

(choose chapter or problem)

For each given path, verify Green's Theorem by showing that

\(\int_{C} y^{2} d x+x^{2} d y=\int_{R} \int\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right) d A\).

For each path, which integral is easier to evaluate? Explain.

(a) C : triangle with vertices (0, 0), (4, 0), (4, 4)

(b) C : circle given by \(x^{2}+y^{2}=1\)

Text Transcription:

int_C y^2 dx+x^2 dy= int_R int(partial N/partial x-partial M/partial y)dA

x^2+y^2=1