Surface Area Of A Surface Of Revolution

Introduction to surface area of a surface of revolution:

A three dimensional shape is obtained by rotating a two dimensional shape around an axis. For example a cylinder is generated by rotating a rectangle around one of its sides. When a right triangle is rotated around any of its legs you get a right circular cone.

This concept with the additional help of calculus helps in deriving the surface area of a surface of revolution.

Surface Area of a Surface of Revolution Derivation of Formula

surface area formula derivation

Look at the above diagram. Let the find surface area formed by the revolution of the surface abcd around x- axis.

Consider an infinitely small strip shown by the segment pqsr within the surface. Being a infinitely small strip, the shape may be considered as a tiny rectangle of height y and width ds.

As per fundamental concept of geometry, the shape generated by this rectangle is a cylinder and its surface area is given by,

dA = 2?y(ds)

The exaggerated figure of the top portion of the rectangle can be computed as,

ds = (dx2 + dy2)1/2 = dx[1 + (dy/dx)2)]1/2 = [1 + (dy/dx)2)]1/2dx

Therefore, dA = 2?y [1 + (dy/dx)2)]1/2dx

Integrating the above between x = a and x = b, gives A, the surface area formed by the revolution of the surface abcd around x- axis.

In other words, A = 2?$int_{a}^{b}sqrt{y [1 + (dy/dx)^2)]}dx$

Surface Area of a Surface of Revolution an Illustration

Let us use the above formula to verify the surface area of a sphere.

surface area - sphere

A sphere is formed by rotating a semicircle around its diameter. Let us assumed a semicircle of radius r with center at origin is rotated about x-axis. The surface area of surface of revolution in this case is the surface area of the sphere.

y = (r2 x2)1/2, dy/dx = -x(r2 x2)-1/2 , (dy/dx) 2 = x2/(r2 x2), [1 + (dy/dx)2)]1/2 = r(r2 x2)-1//2

y[1 + (dy/dx)2)]1/2 = [(r2 x2)1/2][r(r2 x2)-1//2] = r

Using the formula,

A = 2?$int_{a}^{b}sqrt{y [1 + (dy/dx)^2)]}dx$

= 2?r$int_{-r}^{r}dx$ = 2?r(2r) = 4?r2

by: jeri

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