Find the exact length of the curve. y = 5 + 2x3/2, 0 ≤ x ≤ 1
Solution:
Given, function is y = 5 + 2x3/2
Differentiating the function with respect to x,
dy/dx = d(5 + 2x3/2)/dx
dy/dx = 0+ (3 / 2) (2) (x1/2)
dy/dx = 3x1/2 ------------------------ (1)
To find arc length,
Arc length, L = \(\int_{a}^{b}\sqrt{1+(3x^{1/2})^{2}}dx[\)
\(L=\int_{a}^{b}\sqrt{1+9x}dx\)
Now, substitute a = 0 and b = 1
\(L=\int_{0}^{1}\sqrt{1+9x}dx\)
Let, 1+9x = p2
Differentiating with respect to x,
9 dx = 2p dp
dx = 2p / 9 dp
Since we have substituted the function in terms of p, therefore we have to change the limits.
If x = 0 then,
p = √[1+ 9(0)]
p = 1
If x = 1 then,
p = √[1 + 9(1)]
p = √10
Substitute the value of the limits,
\(\\L=\int_{1}^{\sqrt{10}}p^{2}\frac{2}{9}dp \\ \\L=\frac{2}{9}\int_{1}^{\sqrt{10}}p^{2}dp\)
L = 2 / 9 [u3 / 3] in the limit 1 to √10
L = 2 / 9 [10√10 - 1]1/3
L = 2 / 27 (30.623)
L = 2.268 units
Therefore, the length of the curve is 2.268 units.
Find the exact length of the curve. y = 5 + 2x3/2, 0 ≤ x ≤ 1
Summary:
The exact length of the curve y = 5+2x3/2, 0 ≤ x ≤ 1 is 2.268 units.
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