L&T, 19.1-11
If we take area under the curve between and , as above, and then rotate it around the axis through we sweep out a volume called a volume of revolution .
Again divide the area into strips of width . Since the height is , when we rotate the strip we get a disc of radius , see Fig. 6.5 . The area of this disc is , and the volume of the disc is . The total volume is again a sum,
Now take limit where becomes infinitesimal, and thus
This is the formula for the volume of a solid of revolution.
Example 6.3:
Find the volume formed when the curve , between and is rotated around the axis, see Fig. 6.6
Solution:
Example 6.4:
Find the volume formed when equilateral triangle with corners at , , is rotated around the axis, see Fig. 6.7 .
Solution:
Along OA the curve is , along AB the curve is . Thus