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

The exact length of a curve is a geometrical concept that addressed by integral calculus. It is a method for calculating the exact lengths of line segments.

Answer: The exact length of the curve is 4.15.

Let us find the exact length of the curve.

Explanation: 

Given function ⇒ y = 3 + 4x^3/2

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

dy/dx = d(3 + 4x^3/2)/dx

dy/dx = 0 + 4 × (3/2)x^1/2

dy/dx = 6x^(1/2)  ---- (1)

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

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

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

L= \((\int_{0}^{1}\sqrt{1+ 36x}dx\)

Substitute 1 + 36x = z. Then, 36dx = dz

At x = 0, z = 1  and x = 1, z = 37

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

L = \(\int_{1}^{37}\dfrac{z^{1/2}}{36}dz\)

L = \(\left[\dfrac{z^{3/2}}{36.\dfrac{3}{2}}\right]_{1}^{37}\)

L = \(\dfrac{1}{54}[z^{3/2}]_{1}^{37}\)

L = \(\dfrac{1}{54}[37^{3/2} -1^{3/2}]\)

L = \(\dfrac{1}{54}[225.06 - 1]\)

L = 224.06 /54

L = 4.15

Thus, 4.15 is the exact length of the curve.