In this, the fourth section of the course, the topics include:

Interpretations and properties of definite integrals
• Definite integral as a limit of Riemann sums.
• Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval.
• Basic properties of definite integrals (examples include additivity and linearity).

Fundamental Theorem of Calculus
• Use of the Fundamental Theorem to evaluate definite integrals.
• Use of the Fundamental Theorem to represent a particular antiderivative and
the analytical and graphical analysis of functions so defined.

Techniques of antidifferentiation
• Antiderivatives following directly from derivatives of basic functions.
• Antiderivatives by substitution of variables (including change of limits for
definite integrals).

Numerical approximations to definite integrals.
• Use of Riemann sums (using left, right, and midpoint evaluation points) and
trapezoidal sums to approximate definite integrals of functions represented
algebraically, graphically, and by tables of values.


Resources in this section listed by standard.

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Use rectangle approximations to find approximate values of integrals.

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Calculate the values of Riemann Sums over equal subdivisions using left, right, and midpoint evaluation points.

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Interpret a definite integral as a limit of Riemann Sums.

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Understand the Fundamental Theorem of Calculus: Interpret the definite integral of the rate of change of a quantity over an interval as the accumulation of change of the quantity over the interval.

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Use the Fundamental Theorem of Calculus to evaluate definite and indefinite integrals and to represent particular antiderivatives. Perform analytical and graphical analysis of functions so defined.

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Understand and use the properties of definite integrals.

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Understand and use integration by substitution (or change of variable) to find values of integrals.

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Understand and use Riemann Sums and trapezoidal sums, and technology to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values.

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