# Thread: Length of the Curve

1. ## Length of the Curve

Find the exact length of the curve analytically by antidifferentiation. You will need to simplify the integrand algebraically before finding an antiderivative.

$x= \frac{y^3}{3} + \frac{1}{4y}$
from y=1 to y= 3

Hint: $1 + (\frac{dy}{dx})^2)$ is a perfect square.

I don't see the perfect square.

Thanks.

2. Originally Posted by Truthbetold
Find the exact length of the curve analytically by antidifferentiation. You will need to simplify the integrand algebraically before finding an antiderivative.

$x= \frac{y^3}{3} + \frac{1}{4y}$
from y=1 to y= 3

Hint: $1 + (\frac{dy}{dx})^2)$ is a perfect square.

I don't see the perfect square.

Thanks.
$1 = \left ( y^2 - \frac{1}{4y^2} \right ) \frac{dy}{dx}$

$\frac{dy}{dx} = \frac{4y^2}{4y^4 - 1}$

So
$1 + \left ( \frac{dy}{dx} \right )^2$

$= 1 + \left ( \frac{4y^2}{4y^4 - 1} \right )^2$

$= \frac{(4y^4 - 1)^2 + 16y^4}{(4y^4 - 1)^2}$

$= \frac{16y^8 - 8y^4 + 1 + 16y^4}{(4y^4 - 1)^2}$

$= \frac{16y^8 + 8y^4 + 1}{(4y^4 - 1)^2}$

$= \frac{(4y^4 + 1)^2}{(4y^4 - 1)^2}$

-Dan