Question

Double integrals in polar coordinates??
I Know nothing about it :S

Answer 1

Hoot! You just asked your first question! Hang tight while I find people to answer it for you. You can thank people who give you good answers by clicking the 'Good Answer' button on the right!

Answer 2

Sometimes integrals are easier to evaluate using polar
coordinates (r, θ), where x = r cos θ and y = r sin θ.
that is the basic idea behind it

Answer 3

aha,, and how I can apply it ?
can you give me an example ?

Answer 4

I am not that good with it, just looking at my lecture notes, but I can try

Answer 5

The height of the hemisphere about the x-y-plane is given by
z =sqrt(1 − x^2-y^2)
Find the volume under the unit hemisphere with positive z-coordinate and
positive y-coordinate.

Answer 6

region of integration is 1> x^2+y^2 (as it has to be + under sqrt)
and y>=0

Answer 7

now first challenge is to change the region of integration

Answer 8

aha...

Answer 9

x^2+y^2 in polar is =r^2 (as sin^2 +cos^2=1)

Answer 10

so r has to be between 0 and 1

Answer 11

y>0 implies that rsinθ>0
r is always >0 here (0 to 1)
so sinθ has to be >0 implies θ is from 0 to pi

clear so far?

Answer 12

aha ..

Answer 13

\[\int\limits_{0}^{\pi}\int\limits_{0}^{1}\sqrt{1-r ^{2}} drd \theta\]

Answer 14

so this is what we have to do now

Answer 15

it is not, because of the change of integral dxdy becomes r drdθ

Answer 16

\[rsqrt{(1-r ^{2})} \]

this is what you have to integrate

Answer 17

do you know how?

Answer 18

one minute, hoe it comes r from 0 to 1 ,, since r^2<1

Answer 19

how**

Answer 20

1-x^2-y^2 has to be + as it is under sqrt
x^2+y^2= (rcosθ)^2+(rsinθ)^2=r^2
so 1-r^2 has to be +

Answer 21

1-r^2>0
1>r^2
-1<r<1 ??

Answer 22

u are right but the example says:
positive z-coordinate and
positive y-coordinate.

Answer 23

ahaaa
I get it

Answer 24

the answer is 1/3 pi

Answer 25

Thanks alot
I understood the idea

Answer 26

cool!

Answer 27

I have another ques. ?

Answer 28

do you or do you not? :)

Answer 29

go on

Answer 30

in another topic

Answer 31

Im just a 1st year maths student, so my knowledge has limits :)

Answer 32

but I can try

Answer 33

how to find the projection of a=(1,3) on b=(4,0,2) ??

Answer 34

no clue

Answer 35

post it as a different question

Answer 36

ok, thanks for trying :)