Thread: Calculus questions

Originally Posted by ThePerfectHacker

No,

ok, yep i see.

so then if we split the integration into 2 regions, can't we just write this as the answer for part (b)?

Originally Posted by sarahisme

ok, yep i see.

so then if we split the integration into 2 regions, can't we just write this as the answer for part (b)?

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,
if replaced by becomes, so it is symettrical in the xz plane. Thus, that is valid.

Originally Posted by ThePerfectHacker

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,
if replaced by becomes, 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) :




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

Originally Posted by sarahisme

ok, sweet. if the surface above wasn't symmetric, would the answer be this for (b) :

I assume so.

(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

Originally Posted by sarahisme

but when i try to convert to polar coordinates i end up in a mess....

I understand you now.
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.

Thus,

Thus,

Finished. I looks rather simple in polar form.

ok i think i almost have it now....

for part (b) i get:



for (c) i get this answer: