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Subsection 1.2.1 More properties of integration: even and odd functions

Recall  2 We haven't done this in this course, but you should have seen it in your differential calculus course or perhaps even earlier. the following definition

Definition 1.2.9

Let \(f(x)\) be a function. Then,

  • we say that \(f(x)\) is even when \(f(x)=f(-x)\) for all \(x\text{,}\) and
  • we say that \(f(x)\) is odd when \(f(x)=-f(-x)\) for all \(x\text{.}\)

Of course most functions are neither even nor odd, but many of the standard functions you know are.

  • Three examples of even functions are \(f(x)=|x|\text{,}\) \(f(x)=\cos x\) and \(f(x)=x^2\text{.}\) In fact, if \(f(x)\) is any even power of \(x\text{,}\) then \(f(x)\) is an even function.

  • The part of the graph \(y=f(x)\) with \(x\le 0\text{,}\) may be constructed by drawing the part of the graph with \(x\ge 0\) (as in the figure on the left below) and then reflecting it in the \(y\)-axis (as in the figure on the right below).

  • In particular, if \(f(x)\) is an even function and \(a \gt 0\text{,}\) then the two sets
    \begin{align*} &\big\{\ (x,y)\ \big|\ \text{$0\le x\le a$ and $y$ is between $0$ and $f(x)$} \ \big\}\\ &\big\{\ (x,y)\ \big|\ \text{$-a\le x\le 0$ and $y$ is between $0$ and $f(x)$} \ \big\} \end{align*}
    are reflections of each other in the \(y\)-axis and so have the same signed area. That is
    \begin{align*} \int_0^a f(x)\dee{x} &= \int_{-a}^0 f(x)\dee{x} \end{align*}
  • Three examples of odd functions are \(f(x)=\sin x\text{,}\) \(f(x)=\tan x\) and \(f(x)=x^3\text{.}\) In fact, if \(f(x)\) is any odd power of \(x\text{,}\) then \(f(x)\) is an odd function.

  • The part of the graph \(y=f(x)\) with \(x\le 0\text{,}\) may be constructed by drawing the part of the graph with \(x\ge 0\) (like the solid line in the figure on the left below) and then reflecting it in the \(y\)-axis (like the dashed line in the figure on the left below) and then reflecting the result in the \(x\)-axis (i.e. flipping it upside down, like in the figure on the right, below).

  • In particular, if \(f(x)\) is an odd function and \(a \gt 0\text{,}\) then the signed areas of the two sets
    \begin{align*} &\big\{\ (x,y)\ \big|\ \text{$0\le x\le a$ and $y$ is between $0$ and $f(x)$} \ \big\}\\ &\big\{\ (x,y)\ \big|\ \text{$-a\le x\le 0$ and $y$ is between $0$ and $f(x)$} \ \big\} \end{align*}
    are negatives of each other — to get from the first set to the second set, you flip it upside down, in addition to reflecting it in the \(y\)-axis. That is
    \begin{gather*} \int_0^a f(x)\dee{x} = -\int_{-a}^0 f(x)\dee{x} \end{gather*}

We can exploit the symmetries noted in the examples above, namely

\begin{align*} \int_0^a f(x)\dee{x} &= \int_{-a}^0 f(x)\dee{x} & \text{for $f$ even}\\ \int_0^a f(x)\dee{x} &= -\int_{-a}^0 f(x)\dee{x} & \text{for $f$ odd} \end{align*}

together with Theorem 1.2.3 Theorem 1.2.3

\begin{align*} \int_{-a}^a f(x)\dee{x} &= \int_{-a}^0 f(x)\dee{x} + \int_0^a f(x) \dee{x} \end{align*}

in order to simplify the integration of even and odd functions over intervals of the form \([-a,a]\text{.}\)

For any function

\begin{gather*} \int_{-a}^a f(x)\dee{x} = \int_0^a f(x)\dee{x} + \int_{-a}^0 f(x)\dee{x} \end{gather*}

When \(f\) is even, the two terms on the right hand side are equal. When \(f\) is odd, the two terms on the right hand side are negatives of each other.

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