Compare volume formula to integration

[JProgrammer]

So I need to compare the results of the volume formula of a cylinder to the results of the integration.

In geometry, you learn that the volume of a cylinder is given by V = πr2h, where r is the radius and h is the height of the cylinder. Use integration in cylindrical coordinates to confirm the formula V. To do so, let r = 1 and h = 2, and fill in the limits in the integration below. Integrate, then compare the results (one from V and one from integration).

∭1▒〖𝑟 𝑑𝑧 𝑑𝑟 𝑑𝜃〗

The result from the formula is: 2pi

The question I have is what would the cylindrical coordinates be and how would I use them for integration?

 

[Evgeny.Makarov]

The question I have is what would the cylindrical coordinates be

Why don't you look in Wikipedia, MathWorld or the Encyclopedia of Mathematics? See especially the volume element. In fact, you should first make an effort to get a hold of your course materials (a textbook, lecture notes or handouts) because studying them is much more efficient. You should consult online forums for definitions only if it is impossible to obtain them from your instructor or other students.

and how would I use them for integration?

The same as for Cartesian coordinates. You break the volume into small volume elements. Each element is obtained by fixing a point $(\theta,r,z)$ and then varying each coordinate slightly. The resulting volume element is not exactly a cuboid, but can be appriximately considered as such if the coordinate variations are small.

 

[HallsofIvy]

So I need to compare the results of the volume formula of a cylinder to the results of the integration.

In geometry, you learn that the volume of a cylinder is given by V = πr2h, where r is the radius and h is the height of the cylinder. Use integration in cylindrical coordinates to confirm the formula V. To do so, let r = 1 and h = 2, and fill in the limits in the integration below. Integrate, then compare the results (one from V and one from integration).

∭1▒〖𝑟 𝑑𝑧 𝑑𝑟 𝑑𝜃〗

The result from the formula is: 2pi

The question I have is what would the cylindrical coordinates be and how would I use them for integration?

The point is, as the problem said, to 'fill in the limits in the integration below'. "r" is the distance from the central axis to a point in the cylinder. To cover the entire cylinder r will go from the axis. 0, to the cylinder . You are told that the cylinder has radius 1 so your integration, with respect to r, will go from r= 0, at the center, to 1. "z" is measured along the length of the cylinder. You are told that this is h= 2 so to cover the entire cylinder the integration with respect to z is from 0 to h. Finally, $$\theta$$ is the angle measured around the cylinder. To cover the entire cylinder $$\theta$$ must go from 0 to $$2\pi$$. The integral you need to do is
$$\int_{\theta= 0}^{2\pi}\int_{r= 0}^1\int_{z= 0}^2 r dzdrd\theta$$.

What does that integral give you?