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I took Calc I in preparation for the CLEP. I did not learn about solids of revolution. Is this something that is normally part of Calc I and that most Calc II classes will expect that I can do already? I know that not all courses are the same, but is there any general answer for this question?

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  • $\begingroup$ You could read here to see what to expect on the exam: clep.collegeboard.org/science-and-mathematics/calculus $\endgroup$ – Chris Cunningham Oct 6 at 22:20
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    $\begingroup$ Where are you planning to take Calculus II? If it is a college, they will have their topics in a course catalog. It just doesn't seem all that useful to find out if a topic near the borderline is "usually" in one or the other semester of calculus. $\endgroup$ – Chris Cunningham Oct 6 at 22:26
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    $\begingroup$ The answer to the question posed in the title is yes. $\endgroup$ – James S. Cook Oct 6 at 22:34
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    $\begingroup$ It has been in Calc II in the colleges I've worked in, but as others have pointed out, that's no guarantee. I do a bit of it right at the end of Calc I, because I think it makes a great ending for the course. $\endgroup$ – Sue VanHattum Oct 6 at 23:00
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    $\begingroup$ Paul Dawkins at Lamar University labels volumes of revolusion as part of Calculus I. $\endgroup$ – Joseph O'Rourke Oct 6 at 23:31
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As noted by other questions on this website (and in other answers), the course names "Calculus I/II/II/A/B/C/whatever" are not well defined: [1], [2], etc. These are course titles, not mathematical terms. This implies that the curriculum for such a class (or sequences of classes) is also not well defined. Instead, you should look carefully at the syllabi or course descriptions of the courses that you are expecting to take or have taken.

Broadly speaking, undergraduate calculus courses tend to cover the following topics:

  • Limits and Continuity: this typically includes a "naive" approach to limits (with, perhaps, some mention of $\varepsilon$ and $\delta$, though these are not typically covered in much detail) and some basic results on continuity, including the intermediate value theorem;

  • Differential Calculus in One Variable: the definition of the derivative, as well as results for computation (the power rule, the chain rule, L'Hospital's rule, etc.);

  • Integral Calculus in One Variable: the definition of the Riemann integral from Riemann sums, the fundamental theorem of calculus (which links integrals and derivatives), and a god-awful long time on techniques of integration, including "$u$-substitution" (the single variable change-of-variables formula), integration by parts, trigonometric substitution, etc.;

  • Sequences and Series: basic definitions and power series are typically covered here, as well as Taylor's theorem and various tests for convergence;

  • Multivariable Calculus: integration and differentiation in multiple variables—the big results are the change of variables formula and the Green/Gauss/Stokes/Divergence theorems (whatever you want to call these theorems, which are all basically corollaries of each other); and

  • "Applications": various "applications" are typically scattered throughout the curriculum, including models of ballistic flight, some simple differential equations, and (wait for it...) volumes of revolution.

In most of the curricula with which I am familiar, volumes of revolution are typically covered with techniques of integration. In a semester system, they often show up about half-way through the second semester of college calculus, after a bunch of trigonometric substitutions. This might be considered "Calculus II", but might also be called "Calculus IB". On the other hand, in a quarter system, volumes of revolution often show up at the beginning of the third quarter, in what might be called "Calculus IC" or "Calculus III" (depending on the institution).

So, again, read the relevant syllabi or course descriptions. If you see anything like

  • applications (or possibly techniques) of integration, or
  • volumes or solids of revolution,

in the description of the course you intend to take, then you are probaby not expected to know the topic going in. On the other hand, if you see these terms in the descriptions of the prerequisites for the course you intend to take, then you will need to bone up before you enroll.

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There's no clear definition here. When schools find themselves in a situation where they have too much material to put into a single semester of a course, they just split it into two (or more) and add a number to distinguish each section, e.g. Calculus 1 versus Calculus 2 and Algebra 1 versus Algebra 2. These are educational labels, not mathematical ones, so what content falls into each category is going to vary by school.

Personally, I would expect solids of revolution to be in the second semester class and it's far enough in that I would be surprised to see a school fit it into the first semester.

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It is likely that Calculus I, as taught in College, covers volumes of solids of revolution; it may not (or may) be taught in standard Calculus classes (or AP Calulus) in high school. If you hope to take Calculus II in college, after having studying Calculus I in high school, and have not covered volumes of solid of revolution, I'd suggest you look at the following tutorials from Lamar University Calculus I course:

To study these applications of the integral on your own, you can visit Paul's Notes: Calculus I tutorials on two different methods taught: Method of Washers, or rings, and Method of Cylinders.

Also helpful is this tutorial, and Method of Shells.

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    $\begingroup$ To clarify this, volumes of revolution are part of the AP Calculus curriculum for both AB and BC, as an application of integration. In AB, I would expect many syllabuses would have it as the final mini-unit. $\endgroup$ – Matthew Daly Oct 7 at 13:18
  • $\begingroup$ Thanks, @MatthewDaly, for the clarification. I had hoped it would be covered in AP Calc AB and BC. $\endgroup$ – Namaste Oct 7 at 13:24
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I looked at 3 books I had. The two from the 80s (Thomas Finney and Swokowski) seemed to have it in calc 1. Granville (old text) had it in calc 2. The first two had it at chapter 5-6 of 14-15 total (before calc 3 topics). Granville had it chapter 9.

The main differentiator is if you have it right after introduction of simple integration or after learning more elaborate integration techniques. Benefit of the former is more bang for the buck in calc 1 (if never taking calc 2). Presumably the benefit of the latter is more elaborate integrals within the drill.

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