Slope and area

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?

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 $dy/dx$.

For areas, we want $$\int_{x_1}^{x_2} y\; dx = \int_{t_1}^{t_2} y(t) \frac{dx(t)}{dt} dt.$$ To find the area, we need to both compute a derivative and an integral. Here $x_1$ and $t_1$ are the starting values of $x$ and $t$, while $x_2$ and $t_2$ are the ending values.