Using left or right sums, each extra digit of accuracy requires 10 times as many subintervals. Using the trapezoid or midpoint rules, each extra digit of accuracy requires 10 times as many subintervals. If it takes a calculator 4 seconds to evaluate a midpoint approximately accurate to 4 digits, then it will take 40 seconds to increase that approximation to 6 digits. The error in the trapezoid rule is half that of the midpoint rule. 7.8 Improper Integrals 1.
Improper integrals have an infinite limit or an unbounded integrand.
Taking a hydrogen atom apart requires moving the electron to an infinite distance from the proton. An integrand can only become infinite at a one of the limits of integration. An integral that doesn't converge is said to polyverge. 7.9 More on Improper Integrals 1. You may determine that an improper integral converges without knowing its value.