Find the volume of the solid generated by revolving about the x-axis the region bounded by the curve y=x⁴, the line x=1 and the x-axis.

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Answer 1

Revolution around the x-axis means that y forms the radius of revolution and dx is the thickness of the infinitesimal disc that has a volume πy²dx. When y=1, x=1, so x starts at 0 (the vertex of y=x⁴) and moves towards x=1, while radius y goes from 0 to 1. The integral is ₀∫¹(πy²)dx=π₀∫¹x⁸dx=π[x⁹/9]₀¹=π/9 cubic units.