Numerical integration

Two situations force it. Some integrands, $e^{-x^2}$ and $\frac{\sin x}{x}$ among them, have no antiderivative expressible in elementary functions. More often on the exam there is no formula at all, only a table of measurements, and a sum is the only thing you can build. A graph is different: when it is made of line segments and circular arcs, geometry gives the integral exactly, and you only approximate when the curve has no geometric shape.

The four standard estimates all approach the same number as the number of subintervals grows, but not at the same speed. Doubling $n$ roughly halves the error of a left or right sum, and roughly quarters the error of a midpoint or trapezoid estimate, so the second pair is worth using when the data allows it.

On a calculator active question the machine returns the value in one step, but the credit is in the setup. Write the definite integral you evaluated, then the number, rounded to three decimal places or better.