Calculus 7.4 Day 2

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7.4 Day 2 Surface Area
Greg Kelly, Hanford High School, Richland,
Washington
(Photo not taken by Vickie Kelly)
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Surface Area
Consider a curve rotated about the x-axis
The surface area of this band is
The radius is the y-value of the function, so the
whole area is given by
This is the same ds that we had in the length of
curve formula, so the formula becomes
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Surface Area
Difference between surface area and shell
method With surface area, we are integrating
with respect to the change is s (ds) so summing
up the arc length which is the height of the
(shell) only. We find volumes with shell method
because we are integrating the change in x (dx)
so accumulating the thicknesses to find the
volume.
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Example
Rotate about the y-axis.
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Example
Rotate about the y-axis.
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Example
Rotate about the y-axis.
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rotated about x-axis.
Example
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rotated about x-axis.
Example
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Once again
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p
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Example The Area of a Surface of Revolution

• Find the area of the surface formed by revolving
  the graph
• of f(x) x3 on the interval 0, 1 about the
  x-axis, as shown
• in Figure 7.46.

Figure 7.46
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Solution

• The distance between the x-axis and the graph of
  f is r(x) f(x), and because f'(x) 3x2, the
  surface area is

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Example The Area of a Surface of Revolution

• Find the area of the surface formed by revolving
  the graph
• of f(x) x2 on the interval 0, about
  the y-axis, as shown.

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Solution The Area of a Surface of Revolution

• In this case, the distance between the graph of f
  and the y-axis is
• Using the
  surface area is

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Homework7.2 day 2 MMM pgs. 59 62.