Q. 2 - Calculus

Question

The calculus of parametric equations: Let x = f(t) and y = g(t), where f and g are differentiable functions. 

The arc length of the curve defined by the parametric equations on the interval [a, b] is . ....

Step-by-Step Solution

Answer

The arc length is ${\int }_a^b\sqrt{{\left(f\text{'}\left(t\right)\right)}^2+{\left(g\text{'}\left(t\right)\right)}^2}dt$

Step 1. Given information

The functions:

$$\begin{gathered} x=f\left(t\right) \\ y=g\left(t\right) \end{gathered}$$

$a\le t\le b$

Step 2. Arc length of the curve.

The arc length over given interval is:

$$\begin{gathered} l={\int }_a^b\sqrt{{\left(\frac{dx}{dt}\right)}^2+{\left(\frac{dy}{dt}\right)}^2}dt \\ ={\int }_a^b\sqrt{{\left(f\text{'}\left(t\right)\right)}^2+{\left(g\text{'}\left(t\right)\right)}^2}dt \end{gathered}$$