How Is the Surface Area Formula Derived from the Geometry of a Frustum?

[ElijahRockers]

when calculating the surface area of curves rotated about axes, you supposedly integrate 2∏f(x)√(1+f '(x)²) dx. This has been explained to me as being derived from the geometry of a frustum, and I understand how the integration works, but I am confused as the process of deriving this formula - actually churning out the derivation. I feel like most explanations I read on the geometry are skipping a step or two, and it's confusing me.

http://www.analyzemath.com/Geometry/conical_frustum.html

The above link seems to have the most concise method, and I understand how to find all the required measurements in terms of r, R, and H, but they don't actually show how to get the surface area in terms of r, R, and H. I tried a bit of circle geometry to figure it out, but the math started getting pretty hairy.

I figured the surface area of the frustum should be the area of the 'complete' sector minus the area of the 'little' sector. perhaps I still don't completely understand the area of a circle yet?

any insight or links appreciated.

 

[lippyka]

The formula for the surface area of a curve rotated about an axis comes from the equation for the surface area of a conical frustum. A frustum is a cone with two circular bases of different radii. The surface area of a conical frustum is equal to the sum of the areas of both its bases, plus the lateral surface area between the two bases. The lateral surface area can be calculated by integrating the circumference of the circle at any point along the length of the frustum. The circumference of a circle at any point along its length is given by 2πf(x) where f(x) is the radius at that point. Since the radius is changing (as it follows the curve of the frustum), we use the differential form of the circumference equation, which is 2πf(x)√(1+f'(x)2). We then integrate this equation from the lower bound of the frustum (x=a) to the upper bound (x=b) to get the lateral surface area of the frustum. Finally, we add the areas of the two bases to get the total surface area.