How to use the change of variables technique here?

[Dhamnekar Win,odd]

Find the volume  V inside both the sphere x²  + y² + z² = 1 and the cone z = [math]\sqrt{x^2+ y^2}[/math] How to use change of variables technique in this problem?

 

     

[joigus]

Is this homework?

[NTuft]

Itchy trigger finger:

 

I want to integrate by parts...

[mathematic]

Change (x,y) to polar coordinates.

[Dhamnekar Win,odd]

9 hours ago, mathematic said:

Change (x,y) to polar coordinates.

My attempt: I graphed the cone inside the sphere as in my first post. But I don't understand how to use the change of variables technique here to find the required volume. My answer without using integrals is volume of the cone + volume of the spherical cap =

Is this answer correct?

If correct, how to derive this answer using integration technique?

 

 

[Dhamnekar Win,odd]

On 6/2/2022 at 1:09 PM, Dhamnekar Win,odd said:

My attempt: I graphed the cone inside the sphere as in my first post. But I don't understand how to use the change of variables technique here to find the required volume. My answer without using integrals is volume of the cone + volume of the spherical cap =

Is this answer correct?

If correct, how to derive this answer using integration technique?

 

 

 I want to correct the volume asked in the question computed by me = [math] \frac{\pi}{3} \times \frac12 \times \frac{1}{\sqrt{2}} + \frac{\pi}{3} \times (1-\frac{1}{\sqrt{2}})^2 \times (\frac{3}{\sqrt{2}} -(1-\frac{1}{\sqrt{2}}))=0.534497630798[/math] 

[Markus Hanke]

On 6/2/2022 at 2:39 PM, Dhamnekar Win,odd said:

But I don't understand how to use the change of variables technique here to find the required volume.

1. Write down the transformation functions x=..., y=..., z=... for spherical coordinates

2. Calculate the Jacobian matrix from these

3. Calculate the determinant of the Jacobian matrix, in order to get dV=...

4. Determine the integration limits in your new coordinates - radius and equatorial angle are trivial, but for the polar angle you need to find the intersection of the ball and the cone (hint: eliminate z from the equations in the OP)

5. Write down the volume integral using your limits and volume form, and evaluate

[Dhamnekar Win,odd]

5 hours ago, Markus Hanke said:

1. Write down the transformation functions x=..., y=..., z=... for spherical coordinates

2. Calculate the Jacobian matrix from these

3. Calculate the determinant of the Jacobian matrix, in order to get dV=...

4. Determine the integration limits in your new coordinates - radius and equatorial angle are trivial, but for the polar angle you need to find the intersection of the ball and the cone (hint: eliminate z from the equations in the OP)

5. Write down the volume integral using your limits and volume form, and evaluate

I computed the volume inside both the sphere [math] x^2 + y^2 + z^2 =1[/math] and cone [math] z= \sqrt{x^2 + y^2}[/math] as follows:

[math]\displaystyle\int_0^{2\pi} \displaystyle\int_0^{\frac{\pi}{4}}\displaystyle\int_0^1 \rho^2 \sin{\phi}d\rho d\phi d\theta= 0.61343412301 =\frac{(2-\sqrt{2})\pi}{3}[/math]. This answer is correct.

Edited June 3, 2022 by Dhamnekar Win,odd

[Markus Hanke]

Looks correct to me, this is what I got too 👍