The area of the surface generated by revolving about the
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.
Find the area of the surface formed by revolving the astroid about the -axis.
Differentiating the equation of the astroid we get
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
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 .
Now compute the area
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
Compute the area of a surface generated by revolving about the
an arc of the curve
between the points of intersection of the curve and the -axis.
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
Differentiating with respect to
and the linear element