How do you find the volume of the solid generated by revolving the region bounded by the graphs #y=x, y=0, y=4, x=6#, about the line x=6?
1 Answer
Explanation:
Here is the region described:
We can find this area in one of two ways. Method 1 uses intuitive geometric properties to find the volume, while Method 2 uses the Disk Method to find the volume.
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Method 1
Rotating this shape about
In other words, it will be the volume of the cone with base radius
Therefore, as described above, the volume is:
#V = 1/3pi(6)^2(6) - 1/3pi(2)^2(2)#
#V = 1/3pi(216)-1/3pi(8)#
#V = 208/3pi#
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Method 2
Since the axis of rotation is vertical, we will integrate with respect to
The Disk Method formula, therefore, is:
#V = int_a^bpi(x-"axis")^2dy#
#V = piint_0^4(x-6)^2dy#
And since we can make the substitution
#V = piint_0^4(y-6)^2dy#
#V = pi*[(y-6)^3/3]_0^4#
#V = pi[(-2)^3/3 - (-6)^3/3]#
#V = pi[-8/3 + 216/3]#
#V = 208/3pi#