The volume of the resulting solid of revolution about the x-axis will be π/7 cubic units.
When a figure is revolved around some fixed axis, the whole three-dimensional solid formed from the area that it swaps out is called a solid of revolution.
The region under the function over the interval [0,1] on the x-axis revolved about the x-axis.
f(x) = x³
Then the volume of the resulting solid of revolution will be given as
[tex]\rm V = \int _0^1 \pi [f(x)]^2 \ dx[/tex]
Then we have
[tex]\rm V = \int _0^1 \pi (x^3)^2 \ dx\\\\\\V = \pi \int _0^1 x^6 \ dx\\\\\\V = \dfrac{\pi}{7} \left [ x^7 \right ] _0^1\\[/tex]
V = π/7 (1⁷ - 0⁷)
V = π/7 cubic units
Then the volume of the resulting solid of revolution will be π/7 cubic units.
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