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      <text>23.3.1c key point</text>
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      <text><![CDATA[These results are summarized in the following Key Point.
<p></p>
<div class="HELM_keypoint">
<h4>Key Point</h4>
<p>Products of functions</p>
\[
\begin{align}
\text{(even) } \times \text{(even)} &amp;= \text{(even)} \\
\text{(even) } \times \text{(odd)} &amp;= \text{(odd)} \\
\text{(odd) } \times \text{(odd)} &amp;= \text{(even)} 
\end{align}
\]
<p>Sums of functions</p>
\[
\begin{align}
\text{(even) } + \text{(even)} &amp;= \text{(even)} \\
\text{(even) } + \text{(odd)} &amp;= \text{(neither)} \\
\text{(odd) } + \text{(odd)} &amp;= \text{(odd)} 
\end{align}
\]
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Useful properties of even and of odd functions in connection with integrals can be readily deduced if
we recall that a definite integral has the significance of giving us the value of an area as shown in the Figure below.<p><br></p><p style="text-align: center;"><img src="@@PLUGINFILE@@/23-3-1-integral-area.svg" alt="area under a curve" width="201" height="124" class="img-responsive atto_image_button_text-bottom"><br></p><p style="text-align: center;"><br></p>
\( \displaystyle{ \int_a^b } f(t) dt\) gives us the <b>net</b> value of the shaded area, that above the \(t\)-axis being positive, that below being negative.
<p></p>
For the case of a symmetrical interval \( (-a, a)\) for an odd function \(g(t)\) we have 
\[ \displaystyle{\int_{-a} ^a } g(t) dt = 0\]
and for an even function \(h(t)\) we have
\[ \displaystyle{\int_{-a} ^a } h(t) dt = 2 \displaystyle{\int_{0} ^a } h(t) dt \]
This can be seen in the Figure below.<p><br></p><p style="text-align: center;"><img src="@@PLUGINFILE@@/23-3-1-even-odd-integrals.svg" alt="even-odd-integrals" width="440" height="153" class="img-responsive atto_image_button_text-bottom"><br></p>]]></text>
<file name="23-3-1-even-odd-integrals.svg" path="/" encoding="base64">[encoded data, 14092 characters]</file>
<file name="23-3-1-integral-area.svg" path="/" encoding="base64">[encoded data, 9172 characters]</file>
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