Xy coordinates to polar coordinates for double integral. hepl please

[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


thankyou for your time!

 

[gabbagabbahey]

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


thankyou for your time!

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?

 

[Andrew123]

thankyou veeery much!

 

[Mark44]

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)