Volume under a surface which is cut by a circle

In summary, the individual is seeking help with finding the average value of a function over a circle with a radius of 50, centered at z=100, in the z-x plane. The function is only dependent on z and is homogeneous in both the x and y directions. The approach suggested is to use a double integral over a disk in the plane, but the connection between z and x/y is still unclear.
  • #1
Skorpan
2
0
Hi!

I have a function that only depends on z. It is homogeneous in both x and y direction. Then at a certain z-value I need the average value in a circle with radius r from this point. The circle is in the x-plane. How should my integral look like?

Thanks to anyone that can help me.
 
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  • #2
I assume you mean the xy-plane! I'm not clear on what you mean by "at a certain z value" and then have a circle in the xy-plane, with radius r "from this point". How is the point in the xy-plane connected with a z value?

It looks to me like you want the double integral, over a disk in the plane, of f(z). But how is z connected to x and y?
 
  • #3
I have a function f(z). Then i would like to have the average value of this function from a circle in the z-x plane. the centre of the circle should be at let's say z=100 with a radius of 50. so that the values from the function close to z=100 gets a higher importance.
 
  • #4
OH! zx- plane! Now the problem is that it makes no sense to say the center "is at z= 100" since that is a line in the zx-plane, not a point.
 
  • #5
do you know how to find the average value of a function over a domain? cause that sounds exactly like what you want to do
 

1. What is the definition of "Volume under a surface which is cut by a circle"?

The volume under a surface which is cut by a circle refers to the amount of space that is enclosed by the surface and the circle. It represents the three-dimensional measure of the space that is contained within the surface and the circle.

2. How is the volume under a surface which is cut by a circle calculated?

The volume under a surface which is cut by a circle can be calculated by using the formula V = π∫f(x)^2 dx, where f(x) is the function of the surface and the integral is taken over the interval of the circle's diameter. This formula is based on the method of cylindrical shells.

3. What types of surfaces can be cut by a circle to calculate the volume underneath?

Any surface that can be represented by a function can be cut by a circle to calculate the volume underneath. This includes surfaces such as cylinders, spheres, and cones, as well as more complex surfaces defined by mathematical equations.

4. What is the significance of calculating the volume under a surface which is cut by a circle?

Calculating the volume under a surface which is cut by a circle is an important concept in mathematics and physics. It allows us to determine the amount of space enclosed by a given surface, which has practical applications in fields such as engineering, architecture, and fluid mechanics.

5. Are there any real-life examples of the volume under a surface which is cut by a circle being used?

Yes, there are many real-life examples where the concept of volume under a surface which is cut by a circle is used. For instance, in architecture, this concept is used to calculate the volume of a dome or a cylindrical tower. In fluid mechanics, it is used to calculate the volume of a liquid in a cylindrical tank or a pipe. It is also used in calculating the volume of various objects in physics and engineering.

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