The region between y = x^2 and y = 2x is rotated about the x-axis. What is the volume of the solid formed?
Answer and explanation
Correct answer: D. 64π/15 cubic units
- Step 1: Find the intersections: x^2 = 2x gives x = 0 and x = 2, and the line y = 2x lies above the parabola on this interval.
- Step 2: Use the washer method: V = π ∫ from 0 to 2 of [(2x)^2 - (x^2)^2] dx = π ∫ (4x^2 - x^4) dx.
- Step 3: Integrate: π [4x^3/3 - x^5/5] from 0 to 2 = π (32/3 - 32/5) = π (160 - 96)/15 = 64π/15.
Answer
64π/15 cubic units.
Why not the others: '16π/15' squares the difference of the heights instead of subtracting the squares; '32π/3' rotates only the line y = 2x; '128π/15' integrates from -2 to 2 instead of 0 to 2, which doubles the volume.
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