Find the volume of the solid generated by rotating the region bounded by y = sqrt(x), y = 0, and x = 4 around the y-axis.
Answer and explanation
Correct answer: A. 128*pi / 5
- Step 1: Use the method of cylindrical shells: Volume = 2pi int_a^b x * f(x) dx.
- Step 2: Here f(x) = sqrt(x), with limits from x = 0 to x = 4.
- Step 3: Volume = 2pi int_0^4 x x^(1/2) dx = 2pi * int_0^4 x^(3/2) dx.
- Step 4: Anti-derivative of x^(3/2) is (2/5)*x^(5/2).
- Step 5: Evaluate: 2pi (2/5) [4^(5/2) - 0] = (4pi / 5) (32) = 128pi / 5.
Answer
128*pi / 5.
Why not the others: 64pi / 5 forgets the factor of 2 in 2pirh; 32pi / 3 uses disks with incorrect radius bounds; 16pi is an estimate.
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