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Finding the slope of the tangent line to a graph $y=f(x)$ is easy -- just compute $f'(x)$. Likewise, the area under the curve between $x=a$ and $x=b$ is just $\int_a^b f(x) dx$. But how do we compute slopes and areas with parametrized curves? The answer is to express everything in terms of the parameter $t$. By computing derivatives with respect to $t$ and integrals with respect to $t$, we can compute everything we want. For slopes, we are looking for $dy/dx$. This is a limit \begin{eqnarray*} \frac{dy}{dx} &=& \lim \frac{\Delta y}{\Delta x} \cr &=& \lim \frac{\Delta y/\Delta t}{\Delta x/\Delta t} \cr &=& \frac{\lim (\Delta y/\Delta t)}{\lim (\Delta x/\Delta t)} \cr &=& \frac {dy/dt}{dx/dt}, \end{eqnarray*}where the limits are as $\Delta t$ and $\Delta x$ and $\Delta y$ all go to zero. As long as we can take the derivatives of $x$ and $y$, we can compute $dy/dx$.
To find the area under a curve between $x=a$ and $x=b$, we need to compute $$\int_a^b y\; dx = \int_{t_1}^{t_2} y(t) \frac{dx(t)}{dt} dt,$$ where $x(t_1)=a$ and $x(t_2)=b$. To find the area, we need to both compute a derivative as well as an integral.
In the following video, we derive these formulas, work out the tangent line to a cycloid, and compute the area under one span of the cycloid. |