Problem 17 - Entrance Test

Find the length of the curve y = (2/3)x^(3/2) from x = 0 to x = 3.

A. (8/27)(10√10 - 1)B. (8/27)(10√10)C. (10√10 - 1)D. (10√10)

Correct: C

y' = x^(1/2). Arc length: ∫[0,3] √(1 + (y')^2) dx = ∫[0,3] √(1 + x) dx. Let u = 1 + x, du = dx. ∫[1,4] √u du = [(2/3)u^(3/2)] from 1 to 4 = (2/3)(8 - 1) = 14/3. Calculation Error! S = ∫[0, 3] sqrt(1 + x) dx = [2/3 (1 + x)^(3/2)][0, 3] = 2/3 [4^(3/2) - 1^(3/2)] = 2/3(8 - 1) = 14/3