Essential Understanding 3.2
Essential Understanding 3.2 Students will understand that the definite integral of a function over an interval is the limit of a Riemann sum over that interval and can be calculated using a variety of strategies.
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Learning Objective 3.2A(a) Students will be able to interpret the definite integral as the limit of a Riemann sum and (b) express the limit of a Riemann sum in integral notation.
Essential Knowledge 3.2A1 Students will know that a Riemann sum, which requires the 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.
Essential Knowledge 3.2A2 Students will know that the definite integral of a continuous function f, over the interval [a, b], denoted by
, is the limit of Riemann sums as the widths of the subintervals approach 0. That is,
where xi* is a value in the ith subinterval, and Δxi is the width of the ith subinterval, n is the number of subintervals, and max Δxi is the width of the largest subinterval. Another form of the definition is
, where
and xi* is a value in the ith subinterval.
Essential Knowledge 3.2A3 Students will know that the information in a definite integral can be translated into the limit of a related Riemann sum, and the limit of a Riemann sum can be written as a definite integral.
Learning Objective 3.2B Students will be able to approximate a definite integral.
Essential Knowledge 3.2B1 Students will know that definite integrals can be approximated for functions that are represented graphically, numerically, algebraically, and verbally.
Essential Knowledge 3.2B2 Students will know that 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.
Learning Objective 3.2C Students will be able to calculate a definite integral using areas and properties of definite integrals.
Essential Knowledge 3.2C1 Students will know that in some cases, a definite integral can be evaluated by using geometry and the connection between the definite integral and area.
Essential Knowledge 3.2C2 Students will know that properties of definite integrals include the integral of a constant times a function, the integral of the sum of two functions, reversal of limits of integration, and the integral of a function over adjacent intervals.
Essential Knowledge 3.2C3 Students will know that the definition of the definite integral my be extended to functions with removable or jump discontinuities.