Question

Find the exact arc length of the curve y = x^(2/3) over the interval, x = 8 to x = 125 Arc Length = ________

Answer 1

To find the exact arc length of the curve y = x^(2/3) over the interval, x = 8 to x = 125, we can use the formula for arc length and evaluate a definite integral.

Step-by-step explanation:

To find the exact arc length of the curve y = x2/3 over the interval, x = 8 to x = 125, we can use the formula for arc length. The formula for arc length is given by:
Arc Length = ∫√(1 + (dy/dx)²) dx

In this case, dy/dx = (2/3)x^(-1/3). So, we substitute this into the formula:
Arc Length = ∫√(1 + (2/3x^(-1/3))²) dx

We can evaluate this definite integral from x = 8 to x = 125 to find the exact arc length.

Answer 2

The exact value of the arc length of the curve is 118.98 units

How to determine the exact arc length of the curve

From the question, we have the following parameters that can be used in our computation:

`y = x^(2)/(3)`

Also, we have the interval to be

x = 8 to x = 125

The arc length of the curve can be calculated using

`\text{Length} = \int\limits^a_b {\sqrt{1 + ((dy)/(dx))^2}} \, dx`

Recall that

`y = x^(2)/(3)`

So, we have

`(dy)/(dx) = (2)/(3)x^{-(1)/(3)}`

So, we have

`\text{Length} = \int\limits^(125)_(8) {\sqrt{1 + ((2)/(3)x^{-(1)/(3)})^2}} \, dx`

Expand

`\text{Length} = \int\limits^(125)_(8) {\sqrt{1 + (4)/(9)x^{-(2)/(3)}}} \, dx`

Solving further, we have

`\text{Length} = \int\limits^(125)_(8) {\sqrt{\frac{9 + 4x^{-(2)/(3)}}{9}}} \, dx`

Using a graphing tool, we have the integrand to be

`\text{Length} = (\left((4)/(x^(2)/(3))+9\right)^(3)/(2)x)/(27)|\limits^(125)_(8)`

So, we have

`\text{Length} = (\left((4)/(125^(2)/(3))+9\right)^(3)/(2) * 125)/(27) - (\left((4)/(8^(2)/(3))+9\right)^(3)/(2) * 8)/(27)`

Evaluate

Length = 118.98

Hence, the exact arc length of the curve is 118.98 units