Question

Find the length of a curve. y=(4-x^2/3)^3/2 from x=1 to x=8

Answer 1

`y=\left(4-x^(2/3)\right)^(3/2)\implies y'=\frac32\left(4-x^(2/3)\right)^(1/2)\left(-\frac23x^(-1/3)\right)=-(\left(4-x^(2/3)\right)^(1/2))/(x^(1/3))`

The arc length is given by


`\displaystyle\int_1^8\sqrt{1+\left(-(\left(4-x^(2/3)\right)^(1/2))/(x^(1/3))\right)^2}\,\mathrm dx=\int_1^8\sqrt{\frac4{x^(2/3)}}\,\mathrm dx=\int_1^82x^(-1/3)\,\mathrm dx=9`