Overview
We now take a look at types of integrals we have previously stayed away from: An improper integral involves infinity in some way. Amazingly, the seemingly infinite areas (which we should better call “unbounded areas”) involved in these integrals can sometimes be finite. Our study of improper integrals will tie together limits (from Calculus 1) with definite integrals.
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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Identify the two types of improper integrals and state in words how to identify them.
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Given an improper integral with an infinite bound of integration, rewrite it as a proper integral with a limit.
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Given an improper integral with a discontinuous integrand at an upper limit, rewrite it as a proper integral with a limit.
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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Fully evaluate an improper integral and obtain a final value.
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Properly use the words convergent and divergent to describe an improper integral.
To prepare for class
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Do the Preview Activity for this section (on WeBWorK if required by your teacher).
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Watch the following video (by GVSUmath) which explains what improper integrals are in general, and how they are evaluated:
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Watch the following video (by Alan Ableson) which shows an example where an integral with an infinite upper bound must be defined in order to compute the work needed to pull away an object from the Earth’s surface. Watch the video until 4:30, but if you are curious you can watch it until the end if you want to find which (hypothetical) speed an object must have on the surface of the earth in order to escape the gravitational attraction of the Earth (it turns out to be 11 km per second!):
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Watch the following videos (by GVSUmath) which show how to compute an improper integral with an unbounded interval of integration:
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Watch the following videos (by GVSUmath) which show how to compute an improper integral when the integrand has an asymptote at one of the bounds:
