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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.

Let

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).

06_applications_of_the_integral-210.gif

Figure 6.4.10

AREA OF SURFACE OF REVOLUTION

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,

06_applications_of_the_integral-212.gif

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