Surface integral over a cylinder

In summary, the conversation is about a problem that involves finding the volume of a cylinder using integrals and the solution involves using Stokes theorem. The person is confused about how the integral became 36pi, why z is equal to 0, why the double integral over surface S1 is zero, and how the divergence is 3 if z is zero. They also mention that using Stokes theorem does not give the correct answer of 36pi. However, they state that the problem has been solved and they no longer need help.
  • #1
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Homework Statement



Here's a picture of the question:

question5.png


Here's the solution:

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The Attempt at a Solution




I can't really make complete sense of some things around this... Like how did the integral become 36pi based on what is written in the solution? Why is z equal to 0.. What is the double integral over surface S1 and why is it zero? How is the divergence 3 if z is zero? Also if i use stokes theorem and find the volum integral, it would equal to 3 times the volume of the sylinder since the divergence is 3. But it doesn't equal to 36 pi... So if somebody could enlighten me around these questions i'd really appreciate it :)
 
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  • #2
the problem is solved, so no more help is required.
 

1. What is a surface integral over a cylinder?

A surface integral over a cylinder is a mathematical concept that involves calculating the area of a curved surface that wraps around a cylindrical shape. It is a type of double integral that takes into account both the length and circumference of the cylinder.

2. How is a surface integral over a cylinder different from a regular integral?

A surface integral over a cylinder is different from a regular integral because it involves calculating the area of a curved surface, rather than just the area under a curve. It also requires the use of a double integral, whereas a regular integral only uses a single integral.

3. What is the formula for calculating a surface integral over a cylinder?

The formula for calculating a surface integral over a cylinder is ∫∫f(x,y) ds = ∫∫f(x(t),y(t))√(x'(t)^2 + y'(t)^2 + 1) dt, where f(x,y) is the function representing the curved surface, and x(t) and y(t) are the parametric equations for the cylinder.

4. What are some real-world examples of a surface integral over a cylinder?

Some real-world examples of a surface integral over a cylinder include calculating the surface area of a cylindrical soda can, the surface area of a pipe, or the surface area of a cylindrical storage tank.

5. What is the significance of a surface integral over a cylinder in mathematics and science?

A surface integral over a cylinder is significant in mathematics and science because it allows us to calculate the area of curved surfaces, which is important in fields such as physics and engineering. It also provides a way to find solutions to problems involving cylindrical shapes, which are commonly found in real-world applications.

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