# Surface Integral of quarter-cylinder

• misterpickle
In summary, The surface integral over a quarter cylinder involves two different da values, one for the curved outer surface and one for the bottom surface. The da for the curved surface is multiplied by s due to the projection into 3-space. The da for the bottom surface also includes an s term, possibly due to its orientation in the x-y plane. However, the exact reason for this s term is still unclear. Further clarification may be needed.
misterpickle

## Homework Statement

I am taking the surface integral over a quarter cylinder. Everything is fine and I can get the correct answer, it's just a conceptual problem that I need help with.

## Homework Equations

The da for the "curved" outer surface is $$da=sd\phi dz\hat{s}$$
The da for the bottom surface is $$da=sdsd\phi (-\hat{s})$$

I understand why the curved da is multiplied by s, since we are integrating over a surface that is projected into 3-space by a distance s.

I do not understand why this s occurs in the da for the bottom (and top) surface. We integrate over a dynamic ds to find the surface integral for this piece...so why multiply the differential area by s?

The Attempt at a SolutionI think it has something to do with the fact that the top and bottom surfaces of the cylinder are in the x-y plane, so the surface normal is s\hat{s}. But I am still not sure why the s occurs.

## What is the definition of surface integral of quarter-cylinder?

The surface integral of quarter-cylinder is a mathematical concept that measures the flux or flow of a vector field over a curved surface. It involves calculating the dot product of the vector field and the unit normal vector to the surface, then integrating over the surface.

## How is the surface integral of quarter-cylinder calculated?

To calculate the surface integral of quarter-cylinder, the surface is first broken down into small, flat surfaces called infinitesimal elements. The dot product of the vector field and the unit normal vector is then calculated for each element and these values are summed up and multiplied by the surface area of the element. Finally, the sum is integrated over the entire surface.

## What are the applications of surface integral of quarter-cylinder?

The surface integral of quarter-cylinder has various applications in physics, engineering, and mathematics. It is used to calculate the amount of fluid flow through a curved surface, determine the electric flux through a curved surface, and calculate the work done by a force on a curved object, among others.

## What are the differences between surface integral and volume integral?

The main difference between surface integral and volume integral is the shape of the region over which the integration is performed. Surface integrals are calculated over curved surfaces, while volume integrals are calculated over three-dimensional regions. Additionally, surface integrals involve dot products with normal vectors, while volume integrals do not.

## Can the surface integral of quarter-cylinder be extended to other curved surfaces?

Yes, the concept of surface integral can be extended to other curved surfaces, such as spheres, cones, and tori. The method of calculation may vary depending on the shape of the surface, but the underlying principles remain the same.

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