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Surface of Revolution
Motivation:
In the previous section, the length of a curve y=f(x) can be obtained via the integral
We might think a little further: if we rotate the curve along the x-axis, the surface spanned by the length of the curve, denoted as ds , is with area
2 yds and previously we have ds= =
So, the surface area of the infinitesimal ring is
dA = 2 yds = 2 y
To sum up those small pieces of rings, the total area spanned from x=a to x=b is
A = =
This is our “intuitive” result and there could be other ways to find the surface area of a object; or this method might fail to apply on some cases. Thus, we need to think about the condition that every step can hold. The simplest conjecture is that f(x) is differentiable – that can make sure the existence of the derivative function and continuity of the curve.
Theorem ( Surface Area of Revolution ) Let y= f(x) be a differentiable function on [a,b] . Then the surface generated by rotating the curve y= f(x) along x-axis is with area
A=
Example ( Surface Area of Sphere ) E: x2 + y2 = r2 , r > 0 . Find the surface area by rotating E along x-axis.
Sol: It is only necessary consider the effect by rotating y= f(x) =
Then f’(x) = A= = = 4 r2
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