Evaluate the iterated intergal by converting to polar coordinate?

In summary, the conversation is about evaluating an iterated integral by converting it to polar coordinates. The integral is given as \int_0^2\int_0^{\sqrt{2x-x^2}}\sqrt{x^2+y^2} dxdy and the speaker asks for help with converting the limits into polar coordinates. The limits are found to be 0≤r≤2 and 0≤θ≤pi/2.
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  • #2
ZuzooVn said:
Help me Evaluate the iterated intergal by converting to polar coordinate:


http://www.ziddu.com/gallery/4894419/Untitled.jpg.html

hmm … that's [tex]\int_0^2\int_0^{\sqrt{2x-x^2}}\sqrt{x^2+y^2} dxdy[/tex]

ok … I assume you know how to convert dxdy into r and θ

and for the limits, convert y2 ≤ 2x - x2 into r and θ also :wink:
 
  • #3
tiny-tim said:
hmm … that's [tex]\int_0^2\int_0^{\sqrt{2x-x^2}}\sqrt{x^2+y^2} dxdy[/tex]

ok … I assume you know how to convert dxdy into r and θ

and for the limits, convert y2 ≤ 2x - x2 into r and θ also :wink:

0≤y≤1
0≤x≤2
0≤r≤2
0≤θ≤ pi/2
:smile:
 
  • #4
ZuzooVn said:
0≤y≤1
0≤x≤2
0≤r≤2
0≤θ≤ pi/2
:smile:

(have a pi: π :wink:)

No, the upper limit of r will depend on θ.

I repeat … convert y2 ≤ 2x - x2 into r and θ
 

1. What is an iterated integral?

An iterated integral is a type of double or triple integral in which the limits of integration are expressed as a sequence of integrals. It is used to calculate the volume or area of a region bounded by multiple functions.

2. What does it mean to convert to polar coordinates?

Converting to polar coordinates is a way to express a point in the Cartesian coordinate system using a distance (r) and angle (θ) from the origin. It is often used in integration to simplify the calculation of certain types of functions.

3. Why is it useful to evaluate an iterated integral in polar coordinates?

Evaluating an iterated integral in polar coordinates can be useful because it can simplify the calculation of certain types of functions, particularly those with circular or symmetric boundaries. It can also help to visualize and understand the geometric meaning of the integral.

4. How do you convert an iterated integral to polar coordinates?

To convert an iterated integral to polar coordinates, the limits of integration must be changed from rectangular coordinates to polar coordinates. This can be done by substituting the polar coordinate equations (r = √(x^2 + y^2) and θ = tan^-1 (y/x)) into the original integral and changing the differential to account for the change in variable.

5. Are there any limitations to using polar coordinates for evaluating an iterated integral?

While polar coordinates can simplify the calculation of certain types of functions, they may not be suitable for all types of regions or functions. It is important to understand the limitations of polar coordinates and when it is more appropriate to use rectangular coordinates for evaluating an iterated integral.

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