Find the exact length of the curve. y = 5 + 2x^3/2, 0 ≤ x ≤ 1

Solution:

Given, function is y = 5 + 2x^3/2

Differentiating the function with respect to x, 

dy/dx = d(5 + 2x^3/2)/dx

dy/dx = 0+ (3 / 2) (2) (x^1/2)

dy/dx = 3x^1/2 ------------------------ (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.

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Find the exact length of the curve. y = 5 + 2x^3/2, 0 ≤ x ≤ 1

Summary:

The exact length of the curve y = 5+2x^3/2, 0 ≤ x ≤ 1 is 2.268 units.