Question

Find the arc length of the graph of the function over the indicated interval. (Round your answer to three decimal places.) y= (x^5 )/ 10 + 1/(6x^3) [2,5]

Answer 1

The exact value of the arc length of the curve is 123.81 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^5)/(10) + (1)/(6x^3)`

Also, we have the interval to be

x = 2 to x = 5

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^5)/(10) + (1)/(6x^3)`

So, we have

`y = (x^4)/(5) - (1)/(2x^4)`

So, we have

`\text{Length} = \int\limits^(5)_(2) {\sqrt{1 + ((x^4)/(5) - (1)/(2x^4) )^2}} \, dx`

This gives

`\text{Length} = \int\limits^(5)_(2) {\sqrt{1 + ((2x^8 - 5)/(10x^4))^2}} \, dx`

Expand

`\text{Length} = \int\limits^(5)_(2) {\sqrt{1 + (4x^(16) - 20x^6 + 25)/(100x^8)}} \, dx`

Next, we have

`\text{Length} = \int\limits^(5)_(2) {\sqrt{(4x^(16) + 100x^8 - 20x^6 + 25)/(100x^8)}} \, dx`

Using a graphing tool, we have the integrand to be

Length = 123.81

Hence, the exact arc length of the curve is 123.81 units