Find the exact length of the curve. y = 1 + 8x^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: 7.995 is the exact length of the curve.

Explanation: 

Given function ⇒ y = 1 + 8x^3/2

Now, differentiate the given function (1 + 8x^3/2) with respect to “x”

dy/dx = d(1 + 8x^3/2)/dx

dy/dx = 0 + 8 × 3/2 x^1/2

dy/dx = 12x^(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= \(\int_{0}^{1}\sqrt{1+ 144x}dx\)

Substitute 1+144x = z. Then , 144dx = dz

At x = 0, z = 1  and x = 1, z = 144

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

L = \(\int_{1}^{144}\dfrac{z^{1/2}}{144}dz\)

L = \(\left[\dfrac{z^{3/2}}{144.\dfrac{3}{2}}\right]_{1}^{144}\)

L = \(\dfrac{1}{216}[z^{3/2}]_{1}^{144}\)

L= \(\dfrac{1}{216}[144^{3/2} -1^{3/2}]\)

L= \(\dfrac{1}{216}[1728 - 1]\)

L= 1727 /216

L= 7.995

Thus, 7.995 is the exact length of the curve.