- #1

tony873004

Science Advisor

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## Homework Statement

Evaluate the integral [tex]\int {\int\limits_R {\left( {x + y} \right)\,dA} } [/tex] where R is the region that lies to the left of the y-axis between the circles [tex]x^2 + y^2 = 1[/tex] and [tex]x^2 + y^2 = 4[/tex] by changing to polar coordinates.

## Homework Equations

x=r cos theta

y=r sin theta

## The Attempt at a Solution

my effort:

[tex]\begin{array}{l}

r_{inner} = \sqrt 1 = 1,\,\,r_{outer} = \sqrt 4 = 2 \\

R = \left\{ {\left( {r,\theta |1 \le r \le 4,\,\frac{{3\pi }}{2}\,\theta \le \pi } \right)} \right\} \\

x = r\cos \left( \theta \right),\,\,\,y = r\sin \left( \theta \right) \\

\int\limits_1^2 {\int\limits_{\pi /2}^{3\pi /2} {\left( {r\cos \left( \theta \right) + r\sin \left( \theta \right)} \right)\,d\theta \,dr} } \\

\end{array}[/tex]

The solution shows an extra instance of r in the integral. If the original question is for (x+y) then why isn't it simply r cos + r sin, rather than r(r cos + r sin)?