Definite integrals can be approximated using geometric and numerical methods. Approximate a definite integral using geometric and numerical methods. Definite integrals can be approximated for functions that are represented graphically, numerically, analytically, and verbally. Definite integrals can be approximated using a left Riemann sum, a right Riemann sum, a midpoint Riemann sum, or a trapezoidal sum; approximations can be computed using either uniform or nonuniform partitions. Definite integrals can be approximated using numerical methods, with or without technology. Depending on the behavior of a function, it may be possible to determine whether an approximation for a definite integral is an underestimate or overestimate for the value of the definite integral. Interpret the limiting case of the Riemann sum as a definite integral. The limit of an approximating Riemann sum can be interpreted as a definite integral. A Riemann sum, which requires a partition of an interval I, is the sum of products, each of which is the value of the function at a point in a subinterval multiplied by the length of that subinterval of the partition.