Essential Understanding 3.4

Essential Understanding 3.4 Students will understand that the definite integral of a function over an interval is a mathematical tool with many interpretations and applications involving accumulation.

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Learning Objective 3.4A Students will be able to interpret the meaning of a definite integral within a problem.

Essential Knowledge 3.4A1 Students will know that a function defined as an integral represents an accumulation of a rate of change.

Essential Knowledge 3.4A2 Students will know that the definite integral of the rate of change of a quantity over an interval gives the net change of that quantity over that interval.

Essential Knowledge 3.4A3 Students will know that the limit of an approximating Riemann sum can be interpreted as a definite integral.

Learning Objective 3.4B Students will be able to apply definite integrals to problems involving the average value of a function.

Essential Knowledge 3.4B1 Students will know that the average value of a function f over an interval [a, b] is .

Learning Objective 3.4C Students will be able to apply definite integrals to problems involving motion.

Essential Knowledge 3.4C1 Students will know that for a particle in rectilinear motion over an interval of time, the definite integral of velocity represents the particle's displacement over the interval of time, and the definite integral of speed represents the particle's total distance traveled over the interval of time.

Learning Objective 3.4D Students will be able to apply definite integrals to problems involving area and volume.

Essential Knowledge 3.4D1 Students will know that areas of certain regions in the plane can be calculated with definite integrals.

Essential Knowledge 3.4D2 Students will know that volumes of solids with known cross sections, including discs and washers, can be calculated with definite integrals.

Learning Objective 3.4E Students will be able to use the definite integral to solve problems in various contexts.

Essential Knowledge 3.4E1 Students will know that the definite integral can be used to express information about accumulation and net change in many applied contexts.