Category: Mathematics

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Solids of Revolution

Today we saw how to use a definite integral to calculate the volume of a solid of revolution. (Some textbooks will refer to this as finding a “volume of revolution.”)

The solid we studied today is shown below, and the equation used to generate this solid was \(y=\cos x +2\), with \(x\) running from 0 to 5. Can you use your knowledge of solids of revolution to derive the formulas for

  • the volume of a cone with height \(h\) and a base of radius \(r\), \(V=\frac{1}{3}\pi r^2h\)
  • the volume of a sphere of radius \(r\), \(V=\frac{4}{3}\pi r^3\)

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Differentiation and Integration Test

On Sunday, October 16th we’ll have a test on differentiation and integration (not including integration by parts and integration by substitution).

The following questions will be useful in preparing for the test.

(Pearson Textbook)
Pages 751–752—pick any two questions that you have not yet completed
Pages 759–760 2, 3, 13
Pages 762–770 questions 1, 2, 5, 9, 15, 26, 36, 40, 52, 60
Pages 846–853 questions 1, 3, 7,  11 (a and b only), 16

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Integration by Substitution

As discussed on Thursday, try the following questions this weekend. Some may not be as difficult as they first appear, and others…

Complete pages 780–781 questions 26, 28, 33, 45, and 49.
If you’re interested in (what may be) a challenge, also try questions 43, 48, and 50.

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Composites and Inverse Functions Test

On Wednesday, September 28th we’ll have our first HL test on composite and inverse functions.

The following list of questions should be completed as part of your review of this material, and we can discuss any problem you may be having in class before the test.

Pages 85–89 questions 1 to 4, 9 to 11, 15 to 22

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12 HL Composite Functions Homework

Let \(f(x)=x^2\) and \(g(x)=x-1\).

  1. Find the range of \(f\) and \(g\), assuming the domain for both is \(\mathbb{R}\).
  2. Find the range of \(f\) and \(g\), assuming the domain for both is \([-2,\infty[\).
  3. Find the value of each of the functions below when \(x=4\).
    a) \(f\circ g\)
    b) \(g\circ f\)
  4. Find the range of each of the functions in Question 3.