Question

Find the exact length of the curve. y2 = 4(x + 1)3, 0 ≤ x ≤ 1, y > 0

Answer 1

We can find this using the formula: L= ∫√1+ (y')² dx

First we want to solve for y by taking the 1/2 power of both sides:

y=(4(x+1)³)^1/2
y=2(x+1)^3/2

Now, we can take the derivative using the chain rule:

y'=3(x+1)^1/2

We can then square this, so it can be plugged directly into the formula:

(y')²=(3√x+1)²
(y')²=9(x+1)
(y')²=9x+9

We can then plug this into the formula:

L= ∫√1+9x+9 dx *I can't type in the bounds directly on the integral, but the upper bound is 1 and the lower bound is 0
L= ∫(9x+10)^1/2 dx *use u-substitution to solve
L= ∫u^1/2 (du/9)
L= 1/9 ∫u^1/2 du
L= 1/9[(2/3)u^3/2]
L= 2/27 [(9x+10)^3/2] *upper bound is 1 and lower bound is 0
L= 2/27 [19^3/2-10^3/2]
L= 2/27 [√6859 - √1000]
L=3.792318765

The length of the curve is 2/27 [√6859 - √1000] or 3.792318765 units.