Multiple and Line Integrals

Due by Mon, 26 Mar 2018

1. Let $S$ be the unit disc in the plane, i.e., $S = \{(x, y)\in \mathbb{R}^2 : x^2 + y^2 \leq 1\}$. Let $f(x, y) = e^{x^2 + y^2}$.
   • (a) Express $\int_S f(x, y)dxdy$ in polar co-ordinates.
   • (b) Compute this integral by expressing as an iterated integral.
2. Let $C$ be the unit circle traversed in the anti-clockwise direction. Compute the following line integrals along $C$.
   • (a) $\int_C ydx + x dy$.
   • (b) $\int_C ydx - x dy$.
   • (c) $\int_C xdx + y dy$.

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