Integration in polar coordinates

In summary, integration in polar coordinates is a method of finding the area under a curve in a polar coordinate system. It is useful for finding the area of curves that cannot be easily expressed in Cartesian coordinates and for solving problems involving symmetry. The formula for calculating the area is A = ∫1/2r^2dθ, and it can be extended to three-dimensional shapes using multiple integrals. However, it is limited to shapes with radial symmetry and can be more complex than using just Cartesian coordinates for integration.
  • #1
Fernando Revilla
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I quote a question from Yahoo! Answers

By changing to polar coordinates, evaluate the integral.
(Integrand)(integrand)[(x^2+y^2)^(7/2)… where D is the disk x^2+y^2<=16.

I have given a link to the topic there so the OP can see my response.
 
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  • #2
Denote $I=\displaystyle\iint_{D}(x^2+y^2)^{7/2}dxdy$ wiith $D\equiv x^2+y^2\le 16$. We have: $D\equiv \left \{ \begin{matrix}0\le \theta\le 2\pi\\0\le \rho \le 4\end{matrix}\right.$, so $$I=\int_0^{2\pi}d\theta\int_0^4(\rho^2)^{7/2}\rho d\rho=2\pi \left[\frac{\rho^9}{9}\right]_0^4=\frac{2^{19}\pi}{9}$$
 

FAQ: Integration in polar coordinates

What is the definition of integration in polar coordinates?

Integration in polar coordinates is a method of finding the area under a curve in a polar coordinate system. It involves converting the polar coordinates into Cartesian coordinates, using a formula to calculate the area, and then converting back to polar coordinates.

Why is integration in polar coordinates useful?

Integration in polar coordinates is useful because it allows us to find the area of curves that cannot be easily expressed in Cartesian coordinates. It also makes it easier to solve problems involving symmetry, such as finding the area of a sector or the volume of a solid of revolution.

What is the formula for calculating the area using integration in polar coordinates?

The formula for calculating the area using integration in polar coordinates is A = ∫1/2r^2dθ, where r is the distance from the origin to the curve and θ is the angle between the initial line and the curve.

Can integration in polar coordinates be used for three-dimensional shapes?

Yes, integration in polar coordinates can be extended to three-dimensional shapes by using multiple integrals. This allows us to find the volume of a solid with a curved boundary in polar coordinates.

Are there any limitations to using integration in polar coordinates?

One limitation of integration in polar coordinates is that it can only be used for shapes with radial symmetry. Additionally, converting between polar and Cartesian coordinates can be more complex than using just Cartesian coordinates for integration.

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