Approximation of Accumulative Physical Quantity

Similarly to the sum of accumulative physical quantity, physical quantity related to the length of function can also be determine by integration when the physical qunatity can be expressed as infinitesimal elements.

Area of Surface of Revolution

Similar to solid of revolution, a surface of revolution is formed by rotating a curve about an axis. In solid of revolution, the infinitesimal element is approximated by πy2Δx. And the infinitesimal element can also be determined in the same way.

The most simple surface of revolution is the circular cylinderical surface by rotating a horizontal line. The lateral surface area of a circular cylinder can be considered as the extrusion of the circular perimeter along the length of the cylinder. The flat pattern is a rectangular sheet and is equal to 2πrL. Graphically

When the rotating curve is a sloped line with one end on the axis of rotation, the surface of revolution is a right conical surface. Since the slant height is the same for a right lateral conical surface, the flat pattern is a circular sector with radius of the slant height and the arc length of the circular sector is equal to the circumference of the cone base.  Graphically

Mathematically, for the right conical surface, the lateral surface area is

By making use of similar triangles. The lateral surface of a conical frustum is

An infinitesimal lateral surface element is similar to the lateral surface of a conical frustum. Imply

Lateral Surface Areas of Common Shapes

Lateral Surface Area of Cylinder

Lateral surface area of cylinder by integration is

Lateral Surface Area of Cone

Lateral surface area of cone by integration is

Lateral Surface Area of Conical Frustum

Lateral surface area of conical frustum by integration is

Surface Area of Sphere

Surface area of sphere by integration is