Double integral in polar coordinates- funky bounds!?

• May 1st 2011, 03:24 PM
highc1157
Double integral in polar coordinates- funky bounds!?
Hello,

The problem is to convert a cartesian double integral to polar coordinates and evaluate. I'm fine with evaluating, I just require some assistance determining my bounds in polar coordinates :)

double integral f(x,y) = y and order of integration is dy dx
1< x< 2
0 < y < sqrt{4-(x^2)}
• May 1st 2011, 06:26 PM
Prove It
Have you tried making a sketch of this region?
• May 1st 2011, 06:34 PM
Deveno
first of all, you need to determine the limits of your angle θ. since the region of integration is bounded below by the x-axis, one limit for θ is θ = 0.

the second limit will be the angle of the line through the origin and the point where the vertical line x = 1, and the semi-circle y = √(4 - x^2) intersect.

this should not be all that hard for you to deduce (it is a commonly encountered angle). the limits of r will depend on θ, you have two curves:

r = h1(θ), the vertical line x = 1, this will be the hard one to write down, and

r = h2(θ), the semi-circle y = √(4 - x^2). this should be obvious.

writing down f in polar coordinates will not be difficult, but don't forget the adjustment factor for dy dx.
• May 1st 2011, 08:26 PM
highc1157
Quote:

Originally Posted by Prove It
Have you tried making a sketch of this region?

I made a sketch, it looks like the line x=1 makes the region into roughly half of the quarter circle in quadrant one. I know how to find the angles using trig for d\theta , but I cannot figure out the upper and lower radius limits :(
• May 1st 2011, 08:48 PM
Deveno
what is the equation for a circle in polar coordinates? isn't it a really, really simple one?

for example, isn't the circle x^2 + y^2 = 1 just the constant function r = 1 in polar coordinates?

the equation for the vertical line is more challenging. see if you can put the fact that x = rcosθ to good use, if x is constant.
• May 1st 2011, 08:48 PM
Prove It
What is the radius when x = 1 (It will be in terms of $\theta$)? What is the radius when x = 2?