Volumes by Disks

Find the volume of the area between y² = 8x and x = 2 when this area is rotated about the x-axis:

Graph both functions in your calculator or on graph paper to see what the area looks like.
Sketch the figure obtained by rotating the enclosed area around the x-axis.
Draw a representative disk cross section on this solid.
Find an expression for the radius of the disk. r = \/(8x)
To find the total volume, add up all the disk volumes in the solid. (V = p r² h)

Remember, this can be done by using a Riemann Sum or calculating an integral.

2
V(x) = p ò (\/(8x))² dx
0

2
V(x) = p ò 8x dx
0

                            | 2
         =   p (4x²) |       = p (16 - 0) = 16pcubic units
                            | 0

Find the volume of the area between y² = 8x and x = 2 when this area is rotated about the line x = 2:

This time the axis or rotation is a vertical line. The representative disks will be stacked up and down on one another. The radius is y²/8 and the height is dy.

4
V(x) =p ò (y²/8)² dy
-4

                 4                                        | 4                           | 4
V(x) = p ò y⁴/64 dy = (p/64) (y⁵/5) |     =   (p/320) (y⁵) |     =   (p/320) (1024 - (-1024))
            -4                                            | -4                         | -4

= 2048 p
320

= 32p/5 cubic units

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