Greens Theorem: Region with a Hole Let be the region inside the ellipse and outside the

Chapter 15, Problem 44

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Let R be the region inside the ellipse \(x=4 \cos \theta, y=3 \sin \theta\) and outside the circle \(x=2 \cos \theta, y=2 \sin \theta\). Evaluate the line integral

\(\int_{C}\left(3 x^{2} y+1\right) d x+\left(x^{3}+4 x\right) d y\)

where \(C=C_{1}+C_{2}\) is the boundary of R, as shown in the figure.

Text Transcription:

x = 4 cos theta, y = 3 sin theta

x = 2 cos theta, y = 2 sin theta

int_C (3x^2 y + 1) dx + (x^3 + 4x) dy

C = C_1 + C_2