How to use change of variables technique here?

• MHB
• WMDhamnekar
In summary, the conversation is about finding the volume inside both a sphere and a cone using the change of variables technique. The volume can be calculated by expressing the equations in cylindrical coordinates and using the triple integral of the differential of volume.
WMDhamnekar
MHB
Find the volume V inside both the sphere $x^2 + y^2 + z^2 =1$ and cone $z = \sqrt{x^2 + y^2}$

My attempt: I graphed the cone inside the sphere as follows. 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 = $\frac{\pi}{3} \times (\frac12) \times \frac{1}{\sqrt{2}} + \frac{ \pi}{3} *(1-\frac{1}{\sqrt{2}})^2 *(\frac{3}{\sqrt{2}}-(1-\frac{1}{\sqrt{2}}))= 0.497286611528$

Last edited:
I would change to "cylindrical coordinates" in which z is kept and x and y are replaced by polar coordinates r and $\theta$. The boundary equations become $x^2+ y^2+ z^2= r^2+ z^2= 1$ and $z= r$. The "differential of volume" is $r drd\theta dz$. r goes from 0 to 1, $\theta$ goes from 0 to $2\pi$, and z goes from 0 to 1. The volume is given by $\int_{z= 0}^1\int_{\theta= 0}^{2\pi}\int_{r= 0}^1 rdrd\theta dz$

HallsofIvy said:
I would change to "cylindrical coordinates" in which z is kept and x and y are replaced by polar coordinates r and $\theta$. The boundary equations become $x^2+ y^2+ z^2= r^2+ z^2= 1$ and $z= r$. The "differential of volume" is $r drd\theta dz$. r goes from 0 to 1, $\theta$ goes from 0 to $2\pi$, and z goes from 0 to 1. The volume is given by $\int_{z= 0}^1\int_{\theta= 0}^{2\pi}\int_{r= 0}^1 rdrd\theta dz$
This question is 'SOLVED'

Last edited:

1. How do I know when to use the change of variables technique?

The change of variables technique is used when you have a complicated integral that can be simplified by substituting a new variable. This is often the case when you have a function with multiple variables or when the limits of integration are not constants.

2. What are the benefits of using the change of variables technique?

The change of variables technique can simplify a complicated integral and make it easier to evaluate. It can also help to reveal patterns and relationships between variables in the integral.

3. How do I choose the appropriate substitution for the change of variables technique?

The key to choosing the appropriate substitution is to look for a variable or expression that can be substituted in place of the original variable in the integral. This can be determined by looking for patterns, trigonometric functions, or algebraic expressions in the integral.

4. What are some common mistakes to avoid when using the change of variables technique?

One common mistake is forgetting to change the limits of integration when substituting a new variable. It is important to also check for any restrictions on the new variable, such as avoiding division by zero. Another mistake is using an incorrect substitution, which can lead to an incorrect solution.

5. How can I practice and improve my skills in using the change of variables technique?

The best way to practice and improve your skills is by working through various examples and problems that involve the change of variables technique. You can also attend workshops or seek guidance from a mentor or tutor to help you understand the concept better.

• Calculus
Replies
2
Views
805
• Calculus
Replies
29
Views
1K
• Calculus
Replies
4
Views
998
• Calculus
Replies
3
Views
1K
• Calculus
Replies
4
Views
1K
• Calculus
Replies
7
Views
1K
• Calculus
Replies
3
Views
2K
• Calculus
Replies
3
Views
2K
• Calculus
Replies
3
Views
839
• Calculus and Beyond Homework Help
Replies
34
Views
2K