FURTHER APPLICATIONS OF INTEGRATION

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FURTHER APPLICATIONS OF INTEGRATION
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FURTHER APPLICATIONS OF INTEGRATION
8.2Area of a Surface of Revolution
In this section, we will learn about The area of
a surface curved out by a revolving arc.
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SURFACE OF REVOLUTION

• A surface of revolution is formed when a curve
  is rotated about a line.
• Such a surface is the lateral boundary of a solid
  of revolution of the type discussed in Sections
  6.2 and 6.3

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AREA OF A SURFACE OF REVOLUTION

• We want to define the area of a surface of
  revolution in such a way that it corresponds to
  our intuition.
• If the surface area is A, we can imagine that
  painting the surface would require the same
  amount of paint as does a flat region with area
  A .

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AREA OF A SURFACE OF REVOLUTION

• Lets start with some simple surfaces.

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CIRCULAR CYLINDERS

• The lateral surface area of a circular cylinder
  with radius r and height h is taken to be
  A 2prh
• We can imagine cutting the cylinder and
  unrolling it to obtain a rectangle with
  dimensions of 2prh and h.

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CIRCULAR CONES

• We can take a circular cone with base radius r
  and slant height l, cut it along the dashed line
  as shown, and flatten it to form a sector of a
  circle with radius and central angle ? 2pr/l.

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CIRCULAR CONES

• We know that, in general, the area of a sector
  of a circle with radius l and angle ? is ½ l2 ?.

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CIRCULAR CONES

• So, the area is
• Thus, we define the lateral surface area of a
  cone to be A prl.

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AREA OF A SURFACE OF REVOLUTION

• What about more complicated surfaces of
  revolution?

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AREA OF A SURFACE OF REVOLUTION

• If we follow the strategy we used with arc
  length, we can approximate the original curve by
  a polygon.
• When this is rotated about an axis, it creates a
  simpler surface whose surface area approximates
  the actual surface area.
• By taking a limit, we can determine the exact
  surface area.

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BANDS

• Then, the approximating surface consists of a
  number of bandseach formed by rotating a line
  segment about an axis.

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BANDS

• To find the surface area, each of these bands can
  be considered a portion of a circular cone.

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BANDS
Equation 1

• The area of the band (or frustum of a cone) with
  slant height l and upper and lower radii r1 and
  r2 is found by subtracting the areas of two
  cones

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BANDS

• From similar triangles, we have
• This gives

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BANDS
Formula 2

• Putting this in Equation 1, we get
• or
• where r ½(r1 r2) is the average radius of the
  band.

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AREA OF A SURFACE OF REVOLUTION

• Now, we apply this formula to our strategy.

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SURFACE AREA

• Consider the surface shown here.
• It is obtained by rotating the curve y f(x), a
  x b, about the x-axis, where f is positive
  and has a continuous derivative.

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SURFACE AREA

• To define its surface area, we divide the
  interval a, b into n subintervals with
  endpoints x0, x1, . . . , xn and equal width ?x,
  as we did in determining arc length.

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SURFACE AREA

• If yi f(xi), then the point Pi(xi, yi) lies on
  the curve.
• The part of the surface between xi1 and xi is
  approximated by taking the line segment Pi1 Pi
  and rotating it about the x-axis.

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SURFACE AREA

• The result is a band with slant height l
  Pi1Pi and average radius r ½(yi1 yi).
• So, by Formula 2, its surface area is

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SURFACE AREA

• As in the proof of Theorem 2 in Section 8.1, we
  have
• where xi is some number in xi1, xi.

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SURFACE AREA

• When ?x is small, we have yi f(xi) f(xi)
  and yi1 f(xi1) f(xi), since f is
  continuous.
• Therefore,

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SURFACE AREA
Formula 3

• Thus, an approximation to what we think of as
  the area of the complete surface of revolution is

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SURFACE AREA

• The approximation appears to become better as n ?
  8.

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SURFACE AREA

• Then, recognizing Formula 3 as a Riemann sum for
  the function we have

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SURFACE AREADEFINITION
Formula 4

• Thus, in the case where f is positive and has a
  continuous derivative, we define the surface area
  of the surface obtained by rotating the curve y
  f(x), a x b, about the x-axis as

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SURFACE AREA
Formula 5

• With the Leibniz notation for derivatives, this
  formula becomes

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SURFACE AREA
Formula 6

• If the curve is described as x g(y), c y
  d, then the formula for surface area becomes

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SURFACE AREA
Formula 7

• Then, both Formulas 5 and 6 can be summarized
  symbolicallyusing the notation for arc length
  given in Section 8.1as

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SURFACE AREA
Formula 8

• For rotation about the y-axis, the formula
  becomes
• Here, as before, we can use either
• or

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SURFACE AREAFORMULAS

• You can remember these formulas in the following
  ways.

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SURFACE AREAFORMULAS

• Think of 2py as the circumference of a circle
  traced out by the point (x, y) on the curve as
  it is rotated about the x-axis.

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SURFACE AREAFORMULAS

• Think of 2px s the circumference of a circle
  traced out by the point (x, y) on the curve as
  it is rotated about the y-axis.

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SURFACE AREA
Example 1

• The curve , 1 x 1, is an arc of
  the circle x2 y2 4 .
• Find the area of the surface obtained by
  rotating this arc about the x-axis.
• The surface is a portion of a sphere of radius 2.

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SURFACE AREA
Example 1

• We have

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SURFACE AREA
Example 1

• So, by Formula 5, the surface area is

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SURFACE AREA
Example 2

• The arc of the parabola y x2 from (1, 1) to
  (2, 4) is rotated about the y-axis.
• Find the area of the resulting surface.

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SURFACE AREA
E. g. 2Solution 1

• Using y x2 and dy/dx 2x, from Formula 8, we
  have

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SURFACE AREA
E. g. 2Solution 1

• Substituting u 1 4x2, we have du 8x dx.
• Remembering to change the limits of integration,
  we have

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SURFACE AREA
E. g. 2Solution 2

• Using x and dx/dy ,
• we have the following solution.

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SURFACE AREA
E. g. 2Solution 2
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SURFACE AREA
Example 3

• Find the area of the surface generated by
  rotating the curve y ex, 0 x 1, about the
  x-axis.

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SURFACE AREA
Example 3

• Using Formula 5 with y ex and dy/dx ex, we
  have

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SURFACE AREA
Example 3
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SURFACE AREA
Example 3

• Since tan a e , we have
• sec2a 1 tan a 1 e2
• Thus,