Simpson's rule is worth knowing, since it usually gives better results than the midpoint or trapezoidal rule with very little more effort. In practice, between 50 and 100 points will be used, although all features of the method can be illustrated with fewer than 10 points. Separable differential equations are easily solved. Typically, the notation of differentials is used to turn the equation into a couple of integrals that express the general solution when you add C to one of them. You can also find a particular solution that satisfies a given initial condition .
Arc length leads to a special integral.
In the examples used in this course, some trick allows the integral to be evaluated in closed form. Usually, these integrals define new functions. One topic not mentioned that does appear in exercises is the determination of the second derivative of y with respect to x for parametric curves. With a parametric description, derivatives with respect to t are easy to find since everything is expressed in terms of t, so derivatives with respect to x must be found by solving for them in this chain rule equation. Remember to simplify expressions first.