Example 10: Back and forth along a parabola

The parametric curve or trajectory for these parametric equations is a path along the parabola $y=x^2$ :

• 1) from the origin to $(1,1)$ as $t$ varies from 0 to $\pi/2$ .
• 2) from $(1,1)$ back to the origin as $t$ varies from 0 to $\pi$ .
• 3) from the origin to $(-1,1)$ as $t$ varies from $\pi$ to $3\pi/2$ .
• 4) from $(-1,1)$ back to the origin as $t$ varies from $3\pi/2$ to $2\pi$ .

$$\left. \begin{array}{lclcl} x &=& f(t) &=& \sin t\\ y &=& g(t) &=& \sin^2 t\\ \end{array}\right\}\quad 0\leq t\leq 2\pi$$