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Help me Evaluate the iterated intergal by converting to polar coordinate:
http://www.ziddu.com/gallery/4894419/Untitled.jpg.html
http://www.ziddu.com/gallery/4894419/Untitled.jpg.html
ZuzooVn said:Help me Evaluate the iterated intergal by converting to polar coordinate:
http://www.ziddu.com/gallery/4894419/Untitled.jpg.html
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 y^{2} ≤ 2x - x^{2} into r and θ also
ZuzooVn said:0≤y≤1
0≤x≤2
0≤r≤2
0≤θ≤ pi/2
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.
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.
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.
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.
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.