Find the exact length of the curve. y = 3 + 8x^3/2, 0 ≤ x ≤ 1

Solution:

The length of the curve y = f(x) a ≤ x ≤ b is given by:

L = \( \int_{a}^{b}\sqrt{1 + (\frac{\mathrm{d} y}{\mathrm{d} x})^{2}}dx \)

y = 3 + 8x^3/2

dy/dx = 8(3/2)x^1/2

= 12x^1/2

√1 + (dy/dx)² = √1 + (6x^1/2)² = √1 + 144x

Therefore Length of the curve

L = \( \int_{0}^{1}\sqrt{1 + 144x}dx\)

= \( \frac{2}{3}.\frac{1}{144}[(1+144x)^{3/2}]_{0}^{1} \)

= \( \frac{1}{216}[(1+144(1))^{3/2} - (1+144(0))^{3/2})] \)

= \( \frac{1}{216}[(145)^{3/2} - (1)^{3/2})] \)

=\( \frac{1}{216}[(145)^{3/2} - (1)] \)

Find the exact length of the curve. y = 3 + 8x^3/2, 0 ≤ x ≤ 1

Summary:

The length of the curve is L = (1/216)[(145)^3/2 - 1]