Find the exact length of the curve. y = 5 + 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 = 5 + 8x^3/2

Now, we have to differentiate the given function (5 + 8x^3/2) with respect to “x”

dy/dx = d(5 + 8x^3/2)/dx

dy/dx = 0 + 8 × (3/2) x^1/2

dy/dx = 12 x^(1/2) ---- (1)

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

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

Putting the value of dy/dx in the 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 = \(\int_{1}^{144}({z^{(1/2)}}/{144})dz\)

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

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= 1728 /216

L= 7.995

Thus, 7.995 is the exact length of the curve.