Question

Find the length of the curve y = 3/5x^5/3 - 3/4x^1/3 + 6 for 1 < = x < = 8. The length of the curve is . (Type an exact answer, using radicals as needed.)

Answer 1

`\sqrt(387)/(20)`

Explanation:

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

`y=(3)/(5)x^{(5)/(3)}- (3)/(4)x^{(1)/(3)}+6`

`(dy)/(dx) =x^{(2)/(3)}-(1)/(4)x^{-(2)/(3)}`

`1+((dy)/(dx))^2 }=1+(x^{(2)/(3)}-(1)/(4)x^{-(2)/(3)})^2\\=1+(x^{(4)/(3)}-(1)/(2)+ (1)/(16)x^{-(4)/(3)})`

`=(1)/(2)+x^{(4)/(3)}+ (1)/(16)x^{-(4)/(3)}`

For the Interval
`1\leq x\leq 8`

Length of the Curve
`=\int\limits^8_1 {\sqrt{(1)/(2)+x^{(4)/(3)}+ (1)/(16)x^{-(4)/(3)} } } \, dx\\`

Using T1-Calculator

`=\sqrt(387)/(20)`