Find the exact length of the curve: y = 4 + 2x^3/2, 0 ≤ x ≤ 1?

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

Answer: The exact length of the curve y = 4 + 2x^3/2, 0 ≤ x ≤ 1, is 2.2683 units.

Let's solve this question step by step.

Explanation:

Given function is y = 4 + 2x^3/2

Now, we have to differentiate the given function (4 + 2x^3/2) with respect to “x”

dy/dx = d(4 + 2x^3/2)/dx

dy/dx = 0 + (3/2) × 2 × x^1/2

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

To find arc length, we will use the formula for the length of the arc(L), given below:

L = \(\int_{a}^{b}\) √[1 + (dy/dx)²] dx

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

L= \(\int_{a}^{b}\) √[1 + (3x^1/2)²]dx

L= \(\int_{a}^{b}\)√[1 + 9x] dx

Subsitute, a= 0 and b= 1

L=\(\int_{0}^{1}\)√[1 + 9x] dx.

Now, Integrate using substitution method

1+ 9x = u

Differentiate both side w.r.t.x

d(1)/dx + d(9x)/dx = d(u)/dx

dx = du/9

Since we have substituted the function in terms of u, therefore we have to change the limits.

If x = 0 then, u= 1+9(0) = 1

if x = 1 then, u= 1+9(1) = 10

Substitute the values of the limits.

L=\(\int_{1}^{10}\)√[u]/9 du.

⇒ L= (1/9)\(\int_{1}^{10}\)√[u] du

⇒ L= (1/9)\(\int_{1}^{10}\) u^(1/2) du

⇒ L= (1/9)[ u^(3/2) / (3/2) \(]_{1}^{10}\) [Since, \(\int\) xⁿ= (xⁿ⁺¹)/(n +1) ]

⇒ L= (1/9)×(2/3)[ u^(3/2) \(]_{1}^{10}\)

⇒ L= (2/27)[ u^(3/2) \(]_{1}^{10}\)

Put the upper and lower limits.

⇒ L= (2/27)[ 10^(3/2)- 1^(3/2)]

L= 2.2683 units.