CSCE 315 Lecture 21

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Integrals in Polar Coordinates

• Simplify the region
• Simplify the integral

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \iint 4y \, \mathrm{d}A} around a quarter-circle of radius 2 in the first quadrant.

There are three pieces to fix:

1. the range: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \le r \le 2} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \le \theta \le \tfrac{\pi}{2}}
2. The Integrand: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=r\cos{\theta}} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y=r\sin{\theta}}
3. and the differentials: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{d}A=\mathrm{d}x\mathrm{d}y = r\mathrm{d}r\mathrm{d}\theta} . This is because Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} "translates" into Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathrm{d}r} , and $y$ "translates" into $r\mathrm {d} \theta$

Therefore the new integral is "simplified" to

$\int _{0}^{\frac {\pi }{2}}\int _{0}^{2}4r^{2}\sin {\theta }\,\mathrm {d} r\,\mathrm {d} \theta$

Another Example

$\iint _{R}\left(x^{2}-y^{2}\right)\,\mathrm {d} x\,\mathrm {d} y$

1. $x^{2}-y^{2}=r^{2}\cos {(2\theta )}$
2. R: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \le \theta \le \arctan(2)} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 \le r \le \tfrac{2}{\cos{\theta}}}