# Find the exact length of the curve. y = x^{3}/3 + 1/4x , 1 ≤ x ≤ 3

We are going to use using the formula of the exact length of the curve to solve this.

## Answer: The exact length of the curve y = x^{3}/3 + 1/4x, 1 ≤ x ≤ 3 is 9

Let's solve this step by step.

**Explanation:**

Given, y = x^{3}/3 + 1/4x, 1 ≤ x ≤ 3

Length of the curve y = f(x) from x = a to x = b is given by: ∫_{a}^{b} [√1 + {f′(x)}^{2}] dx.

Let's find the first derivative of x.

y = x^{3}/3 + 1/4x

dy/dx = d[x^{3}/3 + 1/4x]/dx

dx/dy = (1/3)×3x^{2} + (1/4)(-1) x^{-2}

dx/dy = x^{2} + (1/4x^{2})

Length of curve = ∫_{1}^{3} √1 + [f′(x)]^{2} dx

= ∫_{1}^{3} √1 + [x^{2} + (1/4x^{2})]^{2} dx

= ∫_{1}^{3} √1 + x^{4} -(1/2)+ (1/16x^{2})] dx

= ∫_{1}^{3} √[x^{2} +(1/2)+ (1/16x^{2})] dx

= ∫_{1}^{3} √[x^{2} + (1/4x^{2})]^{2} dx

= ∫_{1}^{3} [x^{2} + (1/4x^{2})] dx

= [(1/3)x^{3} + (1/4x)]_{1}^{3} dx

= [(1/3)3^{3} + (1/4×3) - (1/3)1^{3} + (1/4×1)]

= [9 + (1/12) - (1/3) + (1/4)]

= 9