Question

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

Answer 1

Answer:

L = 12

Step-by-step explanation:

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

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

Let u = x^2

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