The length of the curve y=x^3 from (0, 0) to (1, 1)

Question

The length of the curve y=x³ from (0, 0) to (1, 1) is

a) 1.732

b) 1.548

c) 1.495

d) 1.414

e) 1.380

Answer 1

Arc length for a function can be defined as follows...

L = int_{lb}^{ub} (sqrt( 1 + (f'(x))² ) *dx

In this case f(x) = x^3 and our lower/upper limits are [0,1] in x

f'(x) = 3x²

(f'(x))² = 9x⁴

sqrt(1 + 9x⁴) is the function which needs to be integrated. This is a complicated integral ( involving the hypergeometric function _nF_m(a;b;c;d) -- in this case --> ₂F₁(-1/2;1/4;5/4,9) ) and as such the approximation using gives --> 1.54787 ~ 1.548 (to three decimal places). The answer is B