Find the exact length of the curve. y = 2 + 8x3/2, 0 ≤ x ≤ 1?
The 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 7.995.
Explanation:
Given function ⇒ y = 2 + 8x3/2
Firstly, we have to differentiate the given function (2 + 8x3/2) with respect to “x”
dy/dx = d(2 + 8x3/2)/dx
dy/dx =
dy/dx = 12 x(1/2) ---- (1)
To find arc length, we use the following formula for the length of the arc(L),
L = \(\int_{x_0}^{x_1}\sqrt{1+ \left(\dfrac{dy}{dx}\right)^2 } dx\)
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.
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