Find the exact length of the curve: y = 5 + 2x3/2, 0 ≤ x ≤ 1?
We will be using the formula of the exact length of the curve and also, use the substitution method to solve this.
Answer: The Exact Length of the Curve y = 5 + 2x3/2, 0 ≤ x ≤ 1 is 2.268 units.
Let's solve this question step by step.
Explanation:
Given function ⇒ y = 5 + 2x3/2
Now, differentiate the given function (5 + 2x3/2) with respect to “x”
dy/dx = d(5 + 2x3/2)/dx
dy/dx = 0 + (3/2) × 2 × x1/2
dy/dx = 3x(1/2) ---- (1)
To find arc length, we use the following formula for the length of the arc(L),
L = \(\int_{a}^{b}\) √1 + [(dy/dx)]2 dx.
Putting the value of dy/dx in length of curve formula from (1)
L= \(\int_{a}^{b}\) √[1 + (3x1/2)]2 dx.
L= \(\int_{a}^{b}\)√[1 + 9x] dx.
Subsitute, a = 0 and b = 1
L=\(\int_{0}^{1}\)√[1 + 9x] dx.
Now, Integrate using substitution method
1+ 9x = u
Differentiate both side w.r.t.x
d(1)/dx + d(9x)/dx = d(u)/dx
dx = du/9
L=\(\int_{0}^{1}\)√[u]/9 du.
Since we have substituted the function in terms of u, therefore we have to change the limits.
If x = 0 then, u= 1+9(0) = 1
if x = 1 then, u= 1+9(1) = 10
Substitute the value the limits.
L=\(\int_{1}^{10}\)√[u]/9 du.
⇒ L= (1/9)\(\int_{1}^{10}\)√[u] du
⇒ L= (1/9)\(\int_{1}^{10}\) u(1/2) du
⇒ L= (1/9)[ u(3/2) / (3/2) \(]_{1}^{10}\) [Since, \(\int\) xn = (xn+1)/(n +1) ]
⇒ L= (1/9)×(2/3)[ u(3/2) \(]_{1}^{10}\)
⇒ L= (2/27)[ u(3/2) \(]_{1}^{10}\)
Put the upper and lower limits.
⇒ L= (2/27)[ 10(3/2) - 1(3/2)]
L= 2.268 units.