You are using symettry, good. But there is one thing which you might not have considered. The regions are symettrical, yes. But what about the surface above it! It happens to be true since,Originally Posted by sarahisme
$\displaystyle f(\sqrt{x^2+y^2}$ if replaced by $\displaystyle -x,-y$ becomes, $\displaystyle f(\sqrt{(-x)^2+(-y)^2}=f(\sqrt{x^2+y^2}$ so it is symettrical in the xz plane. Thus, that is valid.
ok, sweet. if the surface above wasn't symmetric, would the answer be this for (b) :Originally Posted by ThePerfectHacker
as for part (c), i am still struggling with it,
for part (c) i get a graph like this (which is y = +/- sqrt(x-x^2)
and so we want to integrate
0<=x<=1
0<=y<=sqrt(x-x^2)
but when i try to convert to polar coordinates i end up in a mess....
i think thats almost right? :S
I assume so.Originally Posted by sarahisme
(I am just cautioning you, I do not know all the specific rules for manipulating polar coordinates, I might be wrong). For example, when radius can be negative, when it cannot, .... Hate multivarible calculus, so informal )
I do not understand your second problem, rephrase.
as for part (c), i am still struggling with it,
for part (c) i get a graph like this (which is y = +/- sqrt(x-x^2)
and so we want to integrate
0<=x<=1
0<=y<=sqrt(x-x^2)
but when i try to convert to polar coordinates i end up in a mess....
i think thats almost right? :S[/QUOTE]
ok rephrase, umm... i drew this graph
which is a graph of y = +/- (x-x^2)^(1/2)
now i want to integrate over that circle, right? so then i want to integrate over
0<=x<=1
0<=y<=sqrt(x-x^2)
then i want to convert to polar coordinates.
coverting 0<=x<=1 is easy enough, it just gives you 0<=r<=1/cos(theta)
but i can't seem to convert 0<=y<=sqrt(x-x^2) to polar coordinates
I understand you now.Originally Posted by sarahisme
The most efficient way to solve this problem is with a Jacobain substitution. Since the circle is off center you can place it into the center (origon) by a basic substition of x for u+1, that will move the coordinate axes to the left 1 unit and substitute y for v because you are not changing anything.
Doing it your way.
$\displaystyle x^2+y^2=x$
Thus,
$\displaystyle r^2=r\cos \theta$
Thus,
$\displaystyle r=\cos \theta$
Finished. I looks rather simple in polar form.