Question

Common volume for z=3, sphere, x^2+y^2=3*x,cylinder. Use trippel integral and spherical coordinates

Answer 1

x^2-3x+___+y^2=0
x^2-3x+9/4+y^2=9/4
(x-3/2)^2+y^2=9/4
in the xy plane you have a cylinder center at (3/2,0) with radius 3/2
thats going to go up to 3 on the z-axis

i dont know if i am doing this correctly but i dont see a sphere but i can tell you
\[\int\limits_{0}^{\pi/2}\int\limits_{0}^{2\pi}\int\limits_{0}^{3/2}r^2\sin(\Theta)drd (\Theta)d (\phi)\]

Answer 2

i am thinking you need to multiply the intergal by 2 to get the full sphere and not just the top cap, but if its a sphere and z=3 than the spheres radius =3 IMO so on that r=3/2 it could be 3 and the PHI should go from 0topi and then you woulnt have to multiply it by 2.

Answer 3

wrong

Answer 4

are you doing this online or in a book?

Answer 5

a book

Answer 6

ok the only other thing i could think of is
(x-3/2)^2+y^2=z^2/4
that will give you the drawing of a phere

Answer 7

have cal. with cylinder coordinates with correct answer 18pi-24

Answer 8

cylindrical coordinated dzrdrd(theta) is the easy way to do it and spherical coor is just another way

dz=0 to 3 r=0 to 3/2 and d(theta)=0 to 2pi right? thats what you did for this one

Answer 9

no. theta from 0 to pi/2. r from 0 to 3*cos(theta) and z from 0 to sqrt(9-r^2) .
result times 4

Answer 10

ok now it makes sense but when you said x^2+y^2=3*x
im just curious where your getting the 9-r^2 and x^2+y^2=r^2
maybe i didnt read the question in full

Answer 11

phere: x^2+y^2+z^2=3^2 and x^2+y^2=r^2 here you find z

Answer 12

did you mean 3^2 not 3*x in the original eq

Answer 13

no.sphere has r=3

Answer 14

OOO yes than that would have been easier to do, sorry this question took so long, it would have been easier if i would have noticed that in the begining sorry

Answer 15

Ok. No problems, but i wonder about this spherical coord. My problem is to find the correct phi

Answer 16

well i think i can help. so phi starts at xy=0 or on the z axis(positive) and moves towards the xy axis until z=0 and the phi there is pi/2. now thats moving 90 degrees. if you move 180 degrees from the z-axis to the -z-axis the phi is pi.

Answer 17

i found the easiet way to do spereical coords is to to the r first usually going to be something involving (theta) then do the Dtheta next to get rid of the annoying sin and cos and in the end your left with phi which is really easy to plug in

Answer 18

I have found 2 phi's, but when I use them I got 18*pi-27 or 18*pi-36
wrong answer

Answer 19

what were your two phis

Answer 20

sorry. it was cola time.
One phi was pi/2-theta. number two very complicated arccos angle