Calculus-Sec-8-2-Solution

8.2   Area of a Surface of Revolution            by SGLee - HSKim, JYLee, JHLee

1-2. Set up, but do not evaluate, an integral for the area of the surface obtained by rotating the curve about the given axis.

1. -axis

.

2. ;  -axis

(range from a to b) Let,

=>

=>     =>

( or )

3-7. Find the area of the surface obtained by rotating the curve about the -axis.

3. Find the area of the surface when  goes to 0.

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4. Find the area of the surface when  is a positive even integer.

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5.

6.

The surface obtained by rotating the curve about the -axis is

Substitute  ,

=>

Let   and  ,  then

Calculate and simplify to have   .

7. Find the area of the surface obtained by rotating the curve about the -axis.
.

From the above figure, it is clear that the maximum is circle of unit radius in the first quadrant.

When we rotate the unit circle in the first quadrant about the x-axis we obtain half unit sphere. Hence the required area is .

8-10. The given curve is rotated about the -axis. Find the area of the resulting surface.

8.

.

9.

.

10.

The surface obtained by rotating the curve about the -axis becomes

(Range form a to b)

Then substitute  to above equation.

(, then  (because of ) and  so  )

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11. Find the area of the surface of the solid of revolution obtained by rotating about the -axis the circle

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The right semicircle and left semicircle equations are given by

and , respectively.

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12. Use a CAS to find the area of the surface obtained by rotating the curve about the given axis.

Use Simpson's Rule with .

(a) -axis.

(b) -axis