Surface Area, Long Surfaces and Arc Length

Long Surfaces and Arc Length

Let a surface, defined by f(x,y), have a constant cross section for every x. This is like the roof of a barn. It curves up and over in one direction, but is straight along the other.

Apply the plrevious formula for surface area, remembering that the partial with respect to x is 0.

Area = ∫∫sqrt(1 + (f∂y)²)

Integrate with respect to y first, giving the arc length of the cross section of the surface. Then integrate with respect to x, which multiplies the result by the length of the surface. In other words, surface area is arc length across times the transverse length.

The area of a cylinder is easily computed. The top and bottom are circles, having a combined area of 2πr². The wall has a cross sectional arc length of 2πr and a height of h.

Area = 2πr×(r+h)