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# Find the exact length of the curve. y = 5 + 8x^{3/2}, 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: 7.995 is the exact length of the curve.

**Explanation: **

Given function ⇒ y = 5 + 8x^{3/2}

Now, we have to differentiate the given function (5 + 8x^{3/2}) with respect to “x”

dy/dx = d(5 + 8x^{3/2})/dx

dy/dx = 0 + 8 × (3/2) x^{1/2}

dy/dx = 12 x^{(1/2)} ---- (1)

To find arc length, we use the following formula for the length of the arc(L),

L = \(\int_{x_0}^{x_1}\sqrt({1+ \left(dy/dx\right)^2 } )dx\)

Putting the value of dy/dx in the 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 = \(\int_{1}^{144}({z^{(1/2)}}/{144})dz\)

L = \(\left[\dfrac{z^{3/2}}{144 ×(3/2)}\right]_{1}^{144}\)

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= 1728 /216

L= 7.995

### Thus, 7.995 is the exact length of the curve.

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