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Lecture 3

Week # 12 MATH1004

8.3 The Volume of a Solid of Revolution

•The volume of a right circular cylinder is

Volume of Cylinder= π(radius)2(height) = πr2h.

•We can derive the volume of a cylinder with a cylindrical hole, and hole is a cylinder

in its own. Vhole, of the remaining solid which is a shell, is given by

Vhole = 2π(average radius)(width of wall)(height) (1)

where

average radius = rout +rin

2wall width = rout −rin

Examples:

1. Find the volume of the cylindrical solid (of revolution) obtained by rotating the

vertical line segment whose ends are at (x, x2) and (x, 1), and whose width is deﬁned

by the symbol dx, about the y−axis.

2. Find the volume of the cylindrical solid of revolution obtained by rotating the hor-

izontal line segment whose ends are at (0, y) and (√y, y) and whose height is deﬁned

by the symbol dy, about the y−axis, 0 < y < 1.

3. Find the volume of the cylindrical solid of revolution obtained by rotating the ver-

tical line segment whose ends are at (x, 0) and (x, 2x−x2) and whose width is deﬁned

by the symbol dx, about the y−axis. 0 < x < 2

4. Find the volume of the cylindrical solid of revolution obtained by rotating the

vertical line segment whose ends are at (x, 0) and (x, 2x−x2) and whose width is

deﬁned by the symbol dx, about the x−axis.

•Finding the Volume of a Solid of Revolution

1. Sketch the region

2. Decide on a typical slice The rule of thumb here is just lie the one for areas in

the previous section: For rotation about the either the x−axisor y−axis.

iIf it easier to describe the region with function of x, use a vertical slice, otherwise

ii Write all the expressions as functions of y(by ﬁnding the inverse functions) and use

a horizontal slice.

1

3. Find the volume of the slice, dV Use (1).

4. Find the limits f integration These are obtained by ﬁnding the extremities of

the region.

5. Write down the deﬁnite integral for the volume drop all terms containing

the square of either dx or dy from the expression for dV .

6. Evaluate the deﬁnite integral

Examples:

1. Find the volume of the solid of revolution obtained by rotating the region bounded

by the curves y=x2, the y−axis and the line y= 1 about the y−axis.

2. Find the volume of the cylindrical solid of revolution obtained by rotating the region

bounded by the curves y= 2x−x2and y= 0 about y−axis.

3. Find the volume of the cylindrical solid of revolution obtained by rotating the region

bounded by the curves y= 2x−x2and y= 0 about x−axis.

2

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