Overview
In this section we introduce a notation used for expressing the limit of a Riemann sum. We thus define what a definite integral is. We will be discussing the definition, meaning, and use of the definite integral for the remainder of the course. Surprisingly, it has a very strong and natural connection to the derivative — a connection we will discuss and explore in Section 4.4. However, in this section, we deduce the properties that this new mathematical object, the definite integral, has based on its definition and its geometrical interpretation.
Basic Learning Objectives
These are the tasks you should be able to perform with reasonable fluency when you arrive at your next class meeting. Important new vocabulary words are indicated in italics.
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Understand all parts of the limit definition of the definite integral, especially how the definite integral results from taking a limit of Riemann sums.
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Understand what we mean by the integral sign, integrand, and limits of integration, as well as what it means to evaluate a definite integral.
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Understand the geometric interpretation of the quantity denoted by \(\int_a^b f(x) \;dx\).
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Understand some basic key properties that the definite integral possesses, and how we can use these properties to evaluate definite integrals in certain special circumstances.
Advanced Learning Objectives
In addition to mastering the basic objectives, here are the tasks you should be able to perform after class, with practice:
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Understand how \(\int_a^b f(x) \;dx\) can be used to compute the average value of a function on the interval \([a,b]\), and interpret the average value as the height of a rectangle.
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Understand all properties that the definite integral possesses, and how we can use these properties to evaluate definite integrals in certain special circumstances.
To prepare for class
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Watch the following video until 2:10 which establishes the limit of a Riemann sum as a definite integral:
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Watch the following video which shows how we can in simple cases use geometry to compute a definite integral:
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Do the Preview Activity for this section (on WeBWorK if required by your teacher).
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Read subsection 4.3.1 in Active Calculus, which gives the precise definition of the definite integral. Pay special attention to the paragraph discussing the relationship between the definite integral of the velocity and the net change in position, which you should memorize:
$$\displaystyle \int_a^b v(t)\;dt = s(b)-s(a)$$ -
Read subsection 4.3.2 in Active Calculus about properties of the definite integral.
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Watch the end of this video (from 2:10 onwards) which summarizes the properties of the definite integral:
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Watch the following video on how we can use these properties of the integral in practice to quickly compute certain definite integrals:
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Do some experimentation with the following interactive applet (by Marc Renault): Introduction to Integration - Gaining Geometric Intuition
