I have been given the following transformation:
which transforms the circle to the region R in the XY field.
I am requested to find the Volume of R.
I tried using a double integral with polar coordinates, but couldn't exactly understand how i am supposed to use the transformation.
Can someone help me understand what I should be doing here ?
thanks for the reply, but i'm not sure i understood 2 matters:
1. What are the numbers which you used for the double integral ? I noticed that you used: , but what exact values did you extract from that formula ?
2. I need to find the volume of R and not the area. Is there a way to find the volume from the area ?
I don't understand exactly where your trouble is and what exact values you're referring to ......
I'll try to be more clear:
Each of the integrals (within the double integral) should have a lower limit and an upper limit.
I understand that somehow I'm supposed to derive them from the equation: but am not sure how to do this.
In addition, if the answer is , than should I be using polar coordinates ? If so, shouldn't the Jacobian be equal to the circle radius (which is r=1) ?
You don't need to evaluate any integral. We didn't even convert the circle region to polar coords because we simply know what a circle's area is.
I'll try to explain it step by step.
You first have a circle, . Call this region D. The area of this circle is,
Transform this to x-y coordinates, call the new region R. This transformation will give,
Remember, represents the area of a homogenous unit circle (a circle of radius 1). We know that it's .
will be a constant (this is not a general rule but a special case). So we can write,
Note that this Jacobian is different than NonCommAlg's, because I transformed from x-y to u-v while he did it the other way.