# Find the arc length function for the curve with starting point y = 2x^{3/2} with starting point P_{0}(1, 2)

A part of a curve is called an arc and arc length is defined as the distance along the curved line that makes up the arc.

## Answer: The length of the arc is 3.79 units of the given function for the curve.

Let's go through the steps to find the arc length of a given curve.

**Explanation:**

Given function ⇒ y = 2x^{3/2}

Point = P(1, 2)

Firstly, we have to differentiate the given function (y = 2x^{3/2}) with respect to “x”

dy/dx = d(2x^{3/2})/dx

dy/dx = 2(3/2)x^{(1/2)}

dy/dx = 3x^{(1/2)} ------------- (1)

To find arc length, we use the following formula for the length of the arc(L),

L = \(\int_{x_0}^{x_1}\sqrt{(1+ (\dfrac{dy}{dx})^2 )} dx\)

Putting the value of dy/dx in length of curve formula from (1)

L= \((\int_{1}^{2}\sqrt{(1+ (3x^{1/2})^2)}dx\)

L = \(\int_{1}^{2}\sqrt{(1+9x)}dx\)

Substitute 1 + 9x = z and 9x = dz ⇒ dx = dz/9

At x = 1, z = 10 and x = 2, z = 19

Putting the value of “z” and “dz” in the above equation, we get:

L = \(\int_{10}^{19}\dfrac{z^{1/2}}{9}dz\)

L = \(\left[\dfrac{z^{3/2}}{9.\dfrac{3}{2}}\right]_{10}^{19}\)

L = \(\dfrac{2}{27}[z^{3/2}]_{10}^{19}\)

L= \(\dfrac{2}{27}[19^{3/2} -10^{3/2}]\)

L= \(\dfrac{2}{27}[82.82 - 31.62]\)

L= 102.4 / 27

L= 3.79