Volume inside a cone and between z=1 and z=2

In summary, the problem was to evaluate a triple integral in spherical coordinates for the volume inside a cone with given planes. The solution involved substituting values for the spherical coordinates and setting limits for r. However, there was an error in the calculation, as the correct answer should be 7π/3.
  • #1
bigevil
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Homework Statement



Write an evaluate a triple integral in spherical coordinates for the volume inside the cone [tex]z^2 = x^2 + y^2[/tex] between the planes z=1 and z=2.

The answer is 7π/3

The Attempt at a Solution



Substitute values to work out the limits. From [tex]z^2 = x^2 + y^2[/tex], substitute for the spherical coordinates and I get [tex]\theta = \pi/4[/tex]. Similarly substitute [tex]1 = x^2 + y^2[/tex] and [tex]2^2 = x^2 + y^2[/tex], and I find that r ranges from [tex]cosec \theta[/tex] to [tex]2 cosec \theta[/tex].

Then, put together into the triple integral:

[tex]\int_0^{2\pi = \phi} \int_0^{\pi/4} \int_{cosec\theta}^{2cosec\theta} = r^2 sin\theta dr d\theta d\phi = 2\pi \int_0^{\pi/4} \frac{1}{3}(7 cosec^3 \theta)(sin \theta) d\theta = 2\pi (\frac{7}{3})[/tex]

This answer is off. Appreciate if someone could check the working for me, please!
 
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  • #2
bigevil said:
... and I find that r ranges from [tex]cosec \theta[/tex] to [tex]2 cosec \theta[/tex].
Those cosecs should be secs.
 

Related to Volume inside a cone and between z=1 and z=2

1. What is the formula for finding the volume inside a cone?

The formula for finding the volume inside a cone is V = (1/3)πr²h, where r is the radius of the base and h is the height of the cone.

2. How do you calculate the volume between z=1 and z=2?

To calculate the volume between z=1 and z=2, you can use the formula V = (1/3)πr²(h2-h1), where h2 is the height at z=2 and h1 is the height at z=1.

3. Can the volume inside a cone and between z=1 and z=2 be negative?

No, the volume inside a cone and between z=1 and z=2 cannot be negative as volume is a measure of space and cannot have a negative value.

4. How does changing the height of the cone affect the volume between z=1 and z=2?

Changing the height of the cone will affect the volume between z=1 and z=2 by changing the value of h2 and h1 in the formula V = (1/3)πr²(h2-h1). As the height increases, the volume will also increase and vice versa.

5. Can the volume inside a cone and between z=1 and z=2 be calculated without knowing the radius?

Yes, the volume inside a cone and between z=1 and z=2 can be calculated without knowing the radius by using the formula V = (1/3)πh²(z2-z1), where h is the height of the cone and z2 and z1 are the upper and lower limits of the volume.

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