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Solids of revolution

Solids of revolution. When an area is rotated through 2 , a solid object is formed. If a curve is rotated, a hollow object is formed. Both are known as solids of revolution. Volume of revolution about the x-axis.

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Solids of revolution

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  1. Solids of revolution When an area is rotated through 2, a solid object is formed. If a curve is rotated, a hollow object is formed. Both are known as solids of revolution. Volume of revolution about the x-axis The area is bounded by the curve y = f(x) , the x-axis , x = a and x = b, is rotated through 2 radians (360), about the x-axis. The volume of the solid is given by:

  2. Volume of revolution about the x-axis Example Find the volume of the solid formed when the area bounded by y = x3,the x-axis, x = 2 and x = 3, rotated through 2 about the x-axis. Solution

  3. Volume of revolution about the x-axis Example Find the volume of the solid formed when the area bounded by y =(x +1)(x –3)and the x-axis, rotated through 360about the x-axis. Solution

  4. y r x h Volume of revolution about the x-axis Example Prove that the volume of a cone with base r and height h is Solution

  5. y x - r r Volume of revolution about the x-axis Example Prove that the volume of a sphere is Solution

  6. Volume of revolution about the y-axis The area is bounded by the curve y = f(x) , the y-axis , y = c and y = d, is rotated through 2 radians (360), about the y-axis. The volume of the solid is given by:

  7. Volume of revolution about the y-axis Example Find the volume of the solid formed when the area bounded by y = x, the y-axis, y = 1 and y = 2, rotated through 2 about the y-axis. Solution

  8. Volume of revolution about the y-axis Example Find the volume of the solid formed when the area bounded by y = x3, the y-axis, y = 1 and y = 8, rotated through 360 about the y-axis. Solution

  9. Rotating regions between curves If two curves, y = f(x) and y = g(x), intersect at a and b, and f(x) > g(x) in the interval a  x  b, then the volume of the solid of revolution formed by rotating the region between the curves about the x-axis is

  10. Rotating regions between curves Example The area enclosed between the curve y = 4 – x2 and the line y = 4 – 2x is rotated through 360 about the x-axis. Find the volume of the solid generated. Solution 4 – 2x = 4 – x2 x = 0 or x = 2

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