Big Idea 3: Integrals and the Fundamental Theorem of Calculus

Big Idea 3: Integrals and the Fundamental Theorem of Calculus

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Finally we move to our final Big Idea: Integrals. Just like division is the inverse operation to multiplication, integration is the inverse operation to differentiation. Integrals are quite a bit harder than derivatives, but can actually be kinda fun.

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Essential Understanding 3.1 Students will understand that antidifferentiation is the inverse process of differentiation.

Learning Objective 3.1A Students will be able to recognize antiderivatives of basic functions.

Essential Knowledge 3.1A1 Students will know that an antiderivative of a function f is a function g whose derivative is f.

Essential Knowledge 3.1A2 Students will know that differentiation rules provide the foundation for finding antiderivatives.

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.

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

and x_i^* 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 may be extended to functions with removable or jump discontinuities.

Essential Understanding 3.3 Students will understand that the Fundamental Theorem of Calculus, which has two distinct formulations, connects differentiation and integration.

Learning Objective 3.3A Students will be able to analyze functions defined by an integral.

Essential Knowledge 3.3A1 Students will know that the definite integral can be used to define new functions; for example,

Essential Knowledge 3.3A2 Students will know that a if f is a continuous function on the interval [a, b], then , where x is between a and b.

.

Essential Knowledge 3.3A3 Students will know that graphical, numerical, analytical, and verbal representations of a function f provide information about the function g defined as

.

Learning Objective 3.3B(a) Students will be able to calculate antiderivatives and (b) evaluate definite integrals.

Essential Knowledge 3.3B1 Students will know that the function defined by , is an antiderivative of f.

Essential Knowledge 3.3B2 Students will know that if f is continuous on the interval [a, b] and F is an antiderivative of f, then

.

Essential Knowledge 3.3B3 Students will know that the notation means that F'(x) = f(x), and is called an indefinite integral of the function f.

Essential Knowledge 3.3B4 Students will know that many functions do not have closed form antiderivatives.

Essential Knowledge 3.3B5 Techniques for finding antiderivatives include algebraic manipulation (such as long division and completing the square) and substitution of variables.

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.

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.

Essential Understanding 3.5 Students will understand that antidifferentiation is an underlying concept involved in solving separable differential equations. Solving separable differential equations involves determining a function or relation give its rate of change.

Learning Objective 3.5A Students will be able to analyze differential equations to obtain general and specific solutions.

Essential Knowledge 3.5A1 Students will know that antidifferentiation can be used to find specific solutions to differential equations with given initial conditions, including applications to motion along a line, and exponential growth and decay.

Essential Knowledge 3.5A2 Students will know that some differential equations can be solved by separation of variables.

Essential Knowledge 3.5A3 Students will know that solutions to differential equations may be subject to domain restrictions.

Essential Knowledge 3.5A4 Students will know that the function F defined by is a general solution to the differential equation

, and is a particular solution to the equation satisfying F(a) = y₀.

Learning Objective 3.5B Students will be able to analyze differential equations to obtain general and specific solutions.

Essential Knowledge 3.5B1 Students will know that the model for exponential growth that arises from the statement "The rate of change of a quantity is proportional to the size of the quantity" is

.