Calculus/Surface area

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Surface area

Suppose we are given a function and we want to calculate the surface area of the function rotated around a given line. The calculation of surface area of revolution is related to the arc length calculation.

If the function is a straight line, other methods such as surface area formulae for cylinders and conical frustra can be used. However, if is not linear, an integration technique must be used.

Recall the formula for the lateral surface area of a conical frustum:

where is the average radius and is the slant height of the frustum.

For and , we divide into subintervals with equal width and endpoints . We map each point to a conical frustum of width Δx and lateral surface area .

We can estimate the surface area of revolution with the sum

As we divide into smaller and smaller pieces, the estimate gives a better value for the surface area.

Definition (Surface of Revolution)[edit]

The surface area of revolution of the curve about a line for is defined to be

The Surface Area Formula[edit]

Suppose is a continuous function on the interval and represents the distance from to the axis of rotation. Then the lateral surface area of revolution about a line is given by

And in Leibniz notation


As and , we know two things:

  1. the average radius of each conical frustum approaches a single value
  1. the slant height of each conical frustum equals an infitesmal segment of arc length

From the arc length formula discussed in the previous section, we know that


Because of the definition of an integral , we can simplify the sigma operation to an integral.

Or if is in terms of on the interval