Question

The integral \(\frac{1}{{2\pi }}\mathop \int\!\!\!\int \limits_{D\;}^\; \left( {x + y + 10} \right)dxdy\), where D denotes the disc x² + y² ≤ 4, evaluates to__________.

Answer 1

Given,

\(\frac{1}{{2\pi }}\mathop \int\!\!\!\int \limits_{D\;}^\; \left( {x + y + 10} \right)dxdy\)

and x² + y² ≤ 4

Putting x = r.cosθ, y = r.sinθ and dx.dy = r.dr.dθ

\({\rm{I}} = \frac{1}{{2{\rm{\pi }}}}\mathop \smallint \limits_{{\rm{θ }} = 0}^{2{\rm{\pi }}} \mathop \smallint \limits_{{\rm{r}} = 0}^2 \left( {{\rm{rcosθ }} + {\rm{rsinθ }} + 10} \right){\rm{rdrdθ }}\)

\(\\ = \frac{1}{{2{\rm{\pi }}}}\mathop \smallint \limits_{{\rm{θ }} = 0}^{2{\rm{\pi }}} \left( {\frac{{{{\rm{r}}^3}}}{3}{\rm{cosθ }} + \frac{{{{\rm{r}}^3}}}{3}{\rm{sinθ }} + 5{{\rm{r}}^2}} \right)_0^2{\rm{dθ }}\)

\(= \frac{1}{{2{\rm{\pi }}}}\left[ {\mathop \smallint \limits_{{\rm{θ }} = 0}^{2{\rm{\pi }}} \left( {\frac{8}{3}{\rm{cosθ }} + \frac{8}{3}{\rm{sinθ }} + 20} \right){\rm{dθ }}} \right] \)

\(= \frac{1}{{2{\rm{\pi }}}} \left[ {0 + 0 + 20\left( {2{\rm{\pi }}} \right)} \right]= 20\)