Question

Find the length of the smooth arc 1/3(x2 + 2)3/2 + 24 from x=0 to x=3.

Answer 1

L = 12

Explanation:

`\int\limits^a_b {√(1+(f'(x))^2) } \, dx`

`f(x) = ((x^2+2)^3^/^2)/(3)+24\\f'(x)=(3)/(2)*((x^2+2)^1^/^2*2x)/(3)\\ f'(x) = x√(x^2+2)\\ \\\\L = \int\limits^3_0 {\sqrt{1+(x√(x^2+2))^2 } \, dx \\\\\\L = \int\limits^3_0 {√(1+x^2(x^2+2) ) \, dx\\\\\\\\L = \int\limits^3_0 {√(1+x^4+2x^2 ) \, dx\\`

Let u = x^2

`\int\limits^3_0{√(u^2+2u+1) } \, dx \\\\\\\int\limits^3_0 {√((u+1)^2) } \, dx \\\int\limits^3_0 {x^2+1} \, dx \\\\L = x^3/3+x (0to3)\\L = 27/3 + 3\\L = 9 + 3\\L = 12`