Xy coordinates to polar coordinates for double integral. hepl please

• Andrew123
In summary, the region R is converted to { (rcos(@), rsin(@)) | 1 <= r <= 2 , 0 <= @ <= pi/4 } in polar coordinates. The double integral over region R of z=arctan(y/x)dA is evaluated, but there may have been a mistake in the integration. The correct equations for x and y in polar coordinates are r*cos(theta) and r*sin(theta).
Andrew123

Homework Statement

ok change the region R = { (x,y) | 1 <= X^2 + y^2 <= 4 , 0 <= y <= x } to polar region and perform the double integral over region R of z=arctan(y/x)dA

Homework Equations

r^2 = x^2 + y^2, x = r*sin(@), y = r * cos (@)

The Attempt at a Solution

i got R = { (rcos(@), rsin(@) | 1 <= r <= 2 , 0 <= @ <= pi/4 }

and 3/8 * pi ^2 answer in back of book is 3/64 * pi ^2

Andrew123 said:

Homework Statement

ok change the region R = { (x,y) | 1 <= X^2 + y^2 <= 4 , 0 <= y <= x } to polar region and perform the double integral over region R of z=arctan(y/x)dA

Homework Equations

r^2 = x^2 + y^2, x = r*sin(@), y = r * cos (@)

The Attempt at a Solution

i got R = { (rcos(@), rsin(@) | 1 <= r <= 2 , 0 <= @ <= pi/4 }

and 3/8 * pi ^2 answer in back of book is 3/64 * pi ^2

You've correctly converted to polar coordinates and found the limits of integration, but you somehow made a mistake evaluating the integral...Did you by chance forget that you are integrating the function $\tan^{-1}\left(\frac{y}{x}\right)=\theta$ over this region, andf just find the area of the region instead?

thankyou veeery much!

Andrew123;2056564[h2 said:
Homework Equations[/h2]
r^2 = x^2 + y^2, x = r*sin(@), y = r * cos (@)

Not sure this made a difference in your answer, but the equations for x and y above are wrong. They should be
x = r*cos(theta)
y = r*sin(theta)

What are Xy coordinates and polar coordinates?

Xy coordinates are a way of representing points on a two-dimensional plane using two numbers, usually denoted as (x,y). Polar coordinates are another way of representing points on a two-dimensional plane using a distance from the origin (r) and an angle (θ).

How do I convert Xy coordinates to polar coordinates?

To convert Xy coordinates to polar coordinates, you can use the following formulas:
r = √(x² + y²)
θ = tan⁻¹(y/x)

Why would I need to convert Xy coordinates to polar coordinates for a double integral?

In some cases, it may be easier or more efficient to evaluate a double integral using polar coordinates. This is especially true when the region of integration has a circular or radial symmetry.

What is the process for converting a double integral from Xy coordinates to polar coordinates?

To convert a double integral from Xy coordinates to polar coordinates, you will need to change the limits of integration and the integrand. The limits of integration will need to be expressed in terms of r and θ, and the integrand will need to be multiplied by r. You can then evaluate the integral as usual using polar coordinates.

Can you provide an example of converting a double integral from Xy coordinates to polar coordinates?

Yes, for example, if we have the double integral ∫∫R x² + y² dA, where R is the region bounded by the lines x=0, x=1, y=0, and y=1, we can convert this to polar coordinates by changing the limits of integration to 0 to π/4 for θ and 0 to 1 for r, and multiplying the integrand by r. This gives us the integral ∫∫R r² (cos²θ + sin²θ) dr dθ. We can then evaluate this integral using polar coordinates.

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