What is the length of the curve x = y3/2 from y = 0 to y = 4?

Question

What is the length of the curve x = y^3/2 from y = 0 to y = 4?

A	None of the choices below
B
C
D

Answer 1

arc length = ∫_a^b √(1 + f '(x)²) dx

f '(x) = (3/2)x^1/2

Next plug in

arc length = ∫₀⁴ √(1 + ( (3/2)x^1/2 )²) dx = ∫₀⁴ √(1 + (9/4)x) dx

Next make a u substitution:

u = 1 + (9/4)x

du = 9/4 dx, or 4/9du = dx

Your interval of integration now is [1, 10]

∫₁¹⁰ √(u) (4/9) du = (4/9) ∫₁¹⁰ u^1/2du

= (4/9) u^3/2 / (3/2) |₁¹⁰ = (8/27) u^3/2 |₁¹⁰

= (8/27)(10^3/2 - 1^3/2)