# Area of Surface of Revolution

The area of the surface generated by revolving about the -axis the arc of the curve () is expressed by the integral

If a curve is represented parametrically or in polar coordinates, then it is sufficient to change the variable in the above formula, expressing appropriately the differential of the arc length.

## Example 1

Find the area of the surface formed by revolving the astroid about the -axis.

Solution

Differentiating the equation of the astroid we get

whence

Then,

Since the astroid is symmetric about the -axis, in computing the area of the surface we may first assume and then double the result. In other words, the desired area is equal to

Make the substitution

Then

## Example 2

Find the area of the surface generated by revolving about the x-axis a closed contour formed by the curves and

Solution. It is easy to check that the given parabolas intersect at the points and The sought-for area , where the area is formed by revolving the arc , and by revolving the arc .

From the equation we get and .

Hence,

Now compute the area We have and

Thus

## Example 3

Find the area of the surface obtained by revolving a loop of the curve about the -axis.

Solution. The loop is described by a moving point as changes from to Differentiate with respect to both sides of the equation of the curve:

whence Using the formula for computing the area of the surface of a solid of revolution about the -axis, we have

## Example 4

Compute the area of a surface generated by revolving about the -axis an arc of the curve

between the points of intersection of the curve and the -axis.

Solution

Putting we find , and and hence It follows that the curve intersects the -axis at two points: and When is replaced by This curve is thus symmetric about the x-axis.

To find the area of the surface it is sufficient to confine ourselves to the lower portion of the curve that corresponds to the variation of the parameter between and . Differentiating with respect to we find

and the linear element

Hence

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