Setting up a double integral to find the volume

[ahmetbaba]

Homework Statement

Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equation

x²+y²+z²=r²

Homework Equations

Not much equations, just setting the integral up, however I have no idea.

The Attempt at a Solution

I know how to approach these problems if there were only 2 variables, but I'm kind of stuck since there are three variables that I have to deal with.

 

[LCKurtz]

Homework Statement

Set up and evaluate a double integral to find the volume of the solid bounded by the graphs of the equation

x²+y²+z²=r²

Homework Equations

Not much equations, just setting the integral up, however I have no idea.

The Attempt at a Solution

I know how to approach these problems if there were only 2 variables, but I'm kind of stuck since there are three variables that I have to deal with.

Then solve the equation for z, giving one or more functions of two variables.

Is there any obvious symmetry you can use?

 

[ahmetbaba]

how can you solve the equation for z, even then there will be r^2. Help me out with the beginning here please.

 

[LCKurtz]

how can you solve the equation for z, even then there will be r^2. Help me out with the beginning here please.

r is just a constant. Do you recognize what the graph of this is?

 

[ahmetbaba]

do we approach this problem by first saying z=o, then x=o and y=o, integrating all three equations. However the question says to set up a double integration, not a triple integration?

 

[LCKurtz]

do we approach this problem by first saying z=o, then x=o and y=o, integrating all three equations. However the question says to set up a double integration, not a triple integration?

No, you don't do that. So I will ask you again:

1. Do you recognize what this surface is?
2. Can you use any symmetries to your advantage?

Then solve it for z to get started. Get z in terms of x and y if you are required to do a double integral.

 

[ahmetbaba]

well it is a sphere, we can say that the center of the sphere passes through (0,0,0) so if we calculate the top half, we can multiply by 2, to get the answer.

z=sqrt(r^2-x^2-y^2)

the limits being -r and r for the first integral, and sqrt(r^2-y^2) and -sqrt(r^2-y^2)

is this correct so far?

 

[LCKurtz]

well it is a sphere, we can say that the center of the sphere passes through (0,0,0) so if we calculate the top half, we can multiply by 2, to get the answer.

z=sqrt(r^2-x^2-y^2)

the limits being -r and r for the first integral, and sqrt(r^2-y^2) and -sqrt(r^2-y^2)

is this correct so far?

You have the right idea. You might want to change your dxdy integral to polar coordinates to make it easier. If you do that, you might first change the r in the equation of the sphere to a so you don't confuse it with the r in polar coordinates. Good luck. I'm off to bed.

 

[gomunkul51]

x2+y2+z2=r2 is a sphere..
try using spherical coordinates.
set up a triple integral, and do one integration to get to the double integral :)