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Parametic Equations

When a curve is given by parametric equations we get a formula for surface area of revolution analogous to the formula for lengths of parametric curves in Section 6.3.


x = f(t), y = g(t), a ≤ t ≤ h

be a parametric curve in the first quadrant such that the derivatives are continuous and the curve does not retrace its path (Figure 6.4.10).


Figure 6.4.10


06_applications_of_the_integral-211.gif (rotating about y-axis).

To justify this new formula we observe that an infinitesimal piece of the surface is almost a cone frustum of radii x, x + Δx and slant height Δs. Thus compared to Δt,


The Infinite Sum Theorem gives the desired formula for area.

This new formula reduces to our first formula when the curve has the simple form y = f(x). If

y = f(x), a ≤ x ≤ b,

take x = t and get

06_applications_of_the_integral-213.gif (about y-axis).

Similarly, if

x = g(y), a ≤ y ≤ b,

we take y = t and get the formula

06_applications_of_the_integral-214.gif (about y-axis).

Example 3
Example 4

Last Update: 2010-11-25