Calculus Small Changes, Big Ideas

Section 8.3 Trigonometric substitution

• Application: Hydrostatic pressure on a circular drum

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• Emphasize that we’re introducing a more complicated function to simplify the integrand, which is the opposite of the first kind of substitution we did. The rationale is that we’re doing it based on recognizing the format of Pythagorean identities.

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• Make substitutions based on the following:

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  When you see...	Substitute...
  \(a^{2}-x^{2}\)	\(x=a\sin\theta\)
  \(x^{2}+a^{2}\)	\(x=a\tan\theta\)
  \(x^{2}-a^{2}\)	\(x=a\sec\theta\)

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• With definite integrals:

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  \begin{equation*} \int_{a}^{b} f(x)\dd{x}=\int_{g\inv(a)}^{g\inv(b)}f(g(\theta))g'(\theta)\dd{\theta} \end{equation*}

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• Look at examples where symmetry can be used to simplify the limits of integration, such as

  \begin{equation*} \int_{-a}^{a} \sqrt{a^{2}-x^{2}}\dd{x} \end{equation*}

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• If \(f(x)\) is an even function, then

  \begin{equation*} \int_{-a}^{a} f(x)\dd{x}=2\int_{0}^{a} f(x)\dd{x}\text{.} \end{equation*}

  If \(f(x)\) is an odd function, then

  \begin{equation*} \int_{-a}^{a} f(x)\dd{x}=0\text{.} \end{equation*}

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