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{\bf Exam Question

Topic: Double Integral in Polars}

Evaluate the double integral $$\int \!\!\! \int
_R\ln\left(1+x^2+y^2\right)\, d(x,y),$$

where $R$ is the region given by

$$\left\{(x,y)\colon x^2+y^2\le 1\ \ \mathrm{and}\ \ x\le
0\right\}.$$

Given your answer both in terms of $\ln$ and also as a decimal
correct to 3 places.

\vspace{0.5in}

{\bf Solution}

Changing to polar coordinates gives $$I=\int_0^1\,
dr\int_{\pi/2}^{3\pi/2}\ln(1+r^2).r\, d\theta=\pi\int_0^1
r\ln(1+r^2)\, dr.$$ Let $1+r^2=u;\ \ 2rdr=du$
\begin{eqnarray*}
\mathrm{So}\ \ I&=&\pi\int_1^2 \ln u\, du=\frac{\pi}{2}\left[u\ln
u-u\right]_1^2\\&=&\frac{\pi}{2}[2\ln2-2-1\ln1+1]=\frac{\pi}{2}[2\ln2-1]=0.607\
\ \mathrm{(3\ d.p.)} \end{eqnarray*}

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