Change to polar coordinates and integrate

In summary, the problem is asking for the evaluation of an iterated integral through converting to polar coordinates. The given integral is x^2dxdy, with limits of 4 to 0 for the outer integral and √(4y-y^2) to 0 for the inner integral. The attempt at a solution involves converting the limits to polar coordinates and setting up the integral as r^2cos^2(θ), with outer limits from π/2 to 0 and inner limits from 1 to 0. However, the region over which integration takes place is a simple geometric shape, and the set up is not correct.
  • #1
ahmetbaba
23
0

Homework Statement




evaluate the iterated integral by converting to polar coordinates

integral, integral x2dxdy, the limits are 4 to 0 for the outer integral, and /sqrt(4y-y2) to 0 for the inner integral.


Homework Equations





The Attempt at a Solution



well x=rcos(/theta) so x2 = (rcos(/theta))2,

however I have trouble converting the limits to polar coordinates here, I think the outer integrals limits here are from /pi/2 to 0 , and the inner integrals limits are from 1 to 0.

Is this set up correct so far?

So I have, integral, integral r2cos(/theta)2 limits from /pi/2 to 0, and 1 to 0
 
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  • #2
ahmetbaba said:

Homework Statement




evaluate the iterated integral by converting to polar coordinates

integral, integral x2dxdy, the limits are 4 to 0 for the outer integral, and /sqrt(4y-y2) to 0 for the inner integral.


Homework Equations





The Attempt at a Solution



well x=rcos(/theta) so x2 = (rcos(/theta))2,

however I have trouble converting the limits to polar coordinates here, I think the outer integrals limits here are from /pi/2 to 0 , and the inner integrals limits are from 1 to 0.
I don't think so. The region over which integration takes place is a simple geometric shape. What is it?
ahmetbaba said:
Is this set up correct so far?

So I have, integral, integral r2cos(/theta)2 limits from /pi/2 to 0, and 1 to 0
 

FAQ: Change to polar coordinates and integrate

What are polar coordinates?

Polar coordinates are a two-dimensional coordinate system used to represent points in a plane. They use a distance from the origin (r) and an angle from a reference axis (θ) to locate a point.

Why is it necessary to change to polar coordinates when integrating?

Changing to polar coordinates can make certain integrals easier to solve. It can also help in solving problems that involve symmetry or circular shapes.

How do you convert from rectangular coordinates to polar coordinates?

To convert from rectangular coordinates (x, y) to polar coordinates (r, θ), you can use the following equations: r = √(x² + y²) and θ = tan⁻¹(y/x).

What is the process for integrating in polar coordinates?

The process for integrating in polar coordinates involves converting the integrand and the bounds of integration from rectangular coordinates to polar coordinates. Then, using the appropriate integration rules for polar coordinates, the integral can be solved.

When is it most useful to use polar coordinates for integration?

Polar coordinates are most useful for integration when the problem involves circular or symmetric shapes, as well as when the integrand contains terms with powers of x² + y².

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