MAT 21B Lecture Notes - Lecture 11: Jyj
6.4 Areas of Surfaces of Revolution
Surface Area
Suppose a curve is revolved along
an axis and get a surface, what is
the area of that surface?
•Rotate a horizontal line segment (parallel to the
axis of revolution).
We get a cylinder. The surface area is
S= 2πy∆x
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Suppose a curve is revolved along an axis and get a surface, what is the area of that surface: rotate a horizontal line segment (parallel to the axis of revolution). , and s =length of the segment: rotate a curve along an axis. Here: the band width ds is the arc length di erential. That is, ds =pdx2 + dy2 =r( dx dt. )2dt: is the radius from the axis of revolution to an element of arc length ds. In particular, if the curve is given by y = f (x) 0. Then, ds =r1 + ( dy dx and the surface of revolution is. Example: find the area of the surface generated by revolving the curve y = 2 x, 1 x 4, about the x axis. Here, = |x| and the surface of revolution is. In particular, if the curve is given by x = g(y) 0. Then, ds =s( dx dy and the surface of revolution is.
