Surface Area of a Solid of Revolution

rotate

strip location

x coordinates

transparent

strip

views

solid

surface area S ≈ ∑

C

k

×

W

k

,

where

C

k

and

W

k

are the circumference and width of the

th

k

band.

S = ∫2πys

=

b

∫

a

2πy

1+

2

dy

dx

x.

This Demonstration shows the approximation steps that lead to the derivation of the general formula for the surface area of a solid of revolution about the

x

axis:

S=

b

∫

a

2πy

1+

2

dy

dx

.The first view (solid) shows that the surface is a union of bands like the one swept out by the arc PQ. The second view (frustum of a cone) shows how the line segment joining P and Q sweeps out a frustum of a cone. The last view (arc length) shows the dimensions associated with the arc and line segment PQ.