Module 1 : A Crash Course in Vectors
Lecture 2 : Coordinate Systems
  Example 5 :
  Find the integral $\displaystyle{\int e^{-(x^2+y^2)}dxdy}$ where the region of integration is a unit circle about the origin.
  Using polar coordinates the integrand becomes $e^{-\rho^2}$. The range of i integration for $\rho$ is from $0$ to $1$ and for $\theta$ is from $0$ to $2\pi$. The integral is given by
 
\begin{displaymath}I=\int_0^{2\pi}d\theta\int_0^1e^{-\rho^2}\rho d\rho = 2\pi\int_0^1e^{-\rho^2}\rho d\rho\end{displaymath}
  The radial integral is evaluated by substitution $w=\rho^2$ so that $\rho d\rho=dw/2$. The value of the integral is
  \begin{displaymath}I = 2\pi\int_0^1e^{-w}dw/2= \pi[-e^{-w}]_0^1=\pi(1-\frac{1}{e})\end{displaymath}
 


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