Question

(a) How is the length of a curve defined?

(b) Write an expression for the length of a smooth curve given by \(y = f(x)\),\(a \le x \le b.\)

(c) What if\(x\) is given as a function of\(y\) ?

Short answer

a)

The length of a curve is defined to be the limit of sum of the lengths of the segments as the number of points (or the number of segments) increase to infinity.

\(L = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {\left| {{P_{i - 1}}{P_i}} \right|} \)

b)

An expression for the length of a smooth curve is

\(L = \int_a^b {\sqrt {1 + {{\left( {{f^\prime }(x)} \right)}^2}} } dx\)

c)

If\(x\) is given as a function of y then,

\(L = \int_c^d {\sqrt {1 + {{\left( {{g^\prime }(y)} \right)}^2}} } dy\)

Step-by-step solution

Concept used

The length of a curve is defined to be the limit of sum of the lengths of the segments as the number of points (or the number of segments) increase to infinity.

Length of a curve

a)

The length of a curve is defined to be the limit of sum of the lengths of the segments as the number of points (or the number of segments) increase to infinity.

\(L = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {\left| {{P_{i - 1}}{P_i}} \right|} \)

Given information from question

b)

The length of a smooth curve given by \(y = f(x)\),\(a \le x \le b.\)

If \({f^\prime }\) is continuous on the interval (a, b), then the length of the curve \(y = f(x)\) over \(x \in (a,b)\) is given by

\(L = \int_a^b {\sqrt {1 + {{\left( {{f^\prime }(x)} \right)}^2}} } dx.\)

The length of the curve

c)

If\(x\) is given as a function of y then, we use arc length.

If \({g^\prime }\) is continuous on the interval (c, d), then the length of the curve \(x = g(y)\) over \(y \in (c,d)\) is given by

\(L = \int_c^d {\sqrt {1 + {{\left( {{g^\prime }(y)} \right)}^2}} } dy\)