What is the length of the arc of the curve y = (2/3) x^(3/2) from x = 0 to x = 3?
Answer and explanation
Correct answer: A. 14/3 units
- Step 1: Arc length formula: L = integral from a to b of sqrt(1 + (dy/dx)^2) dx.
- Step 2: Compute derivative: dy/dx = (2/3) (3/2) x^(1/2) = sqrt(x).
- Step 3: Square derivative: (dy/dx)^2 = x.
- Step 4: Integrand: sqrt(1 + x).
- Step 5: Integrate: integral from 0 to 3 of (1 + x)^(1/2) dx = [(2/3) * (1 + x)^(3/2)] from 0 to 3.
- Step 6: Evaluate: (2/3) [(4)^(3/2) - (1)^(3/2)] = (2/3) [8 - 1] = (2/3) * 7 = 14/3 units.
Answer
14/3 units.
Why not the others: '7/3 units' forgets the coefficient 2 in 2/3; '8 units' evaluates 4^(3/2) alone; '16/3 units' forgets to subtract the lower limit at x = 0.
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