Find the exact length of the curve: y = 3 + 4x3/2, 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 + 4x3/2
Now, differentiate the given function (3 + 4x3/2) with respect to “x”
dy/dx = d(3 + 4x3/2)/dx
dy/dx = 0 + 4 × (3/2)x1/2
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