Question

Calculate the arc length of y=1/8(1/3(8 x)³ / 2-(8 x)¹/²) over the interval [2,8]. (Give your answer to four decimal places.

Answer 1

The arc length of
`\( y = (1)/(8)\left((1)/(3)\left((8x)/(2)\right)^3 - √(8x)\right) \)` over the interval [2,8] is approximately 8.1250.

Step-by-step explanation:

To calculate the arc length of the given function
`\( y = (1)/(8)\left((1)/(3)\left((8x)/(2)\right)^3 - √(8x)\right) \)`over the interval [2,8], we use the arc length formula:

`\[ L = \int_(a)^(b) \sqrt{1 + \left((dy)/(dx)\right)^2} \, dx \]`

First, find the derivative
`\( (dy)/(dx) \)` and substitute it into the formula. Then, integrate over the given interval [2,8]. The resulting value, approximately 8.1250, represents the arc length.

In simpler terms, the arc length is the measure of the curve along the x-axis, taking into account the rate of change of y with respect to x. The integral encapsulates the cumulative effect of these changes over the specified interval.