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      <text>27.4.1.1 Changing Variables in Multiple Integrals</text>
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      <text><![CDATA[<h3>Changing variables in multiple integrals</h3>
<p>When the method of substitution is used to solve an integral of the form \(\displaystyle\int_{a}^{b}f(x)\;\mathrm{d}x\) three parts of the integral are changed, the limits, the function and the infinitesimal \(\mathrm{d} x\). So if the substitution is of the form \(x=x(u)\) the \(u\) limits, \(c\) and \(d\), are found by solving \(a=x(c)\) and \(b=x(d)\) and the function is expressed in terms of \(u\) as \(f(x(u))\). </p>
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<img src="@@PLUGINFILE@@/Fig364_5.svg" alt="graph showing the intervals delta x and delta y on a curve.">
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<p>The figure shows why the \(\mathrm{d} x\) needs to be changed. While the \(\delta u\) is the same length for all \(u\), the \(\delta x\) change as \(u\) changes. The rate at which they change is precisely \(\displaystyle \frac{\mathrm{d}}{\; \mathrm{d}u}x(u)\). This gives the relation \(\displaystyle \delta x = \frac{\mathrm{d} x}{\; \mathrm{d}u}\delta u\).
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Hence the transformed integral can be written as \[\displaystyle \displaystyle\int_{a}^{b}f(x)\;\mathrm{d}x =\displaystyle\int_{c}^{d}f\left(x\left(u\right)\right)\frac{\mathrm{d} x}{\mathrm{d}u} \mathrm{d}u.\]
Here the \(\displaystyle \frac{\mathrm{d} x}{\mathrm{d}u}\) is playing the part of the Jacobian that we will define.</p>
<p>Another change of coordinates that you have seen is the transformations from cartesian coordinates \((x,y)\) to polar coordinates \((r,\theta)\).<br>Recall that a double integral in polar coordinates is expressed as \(\qquad\displaystyle \displaystyle\int\!\!\displaystyle\int f(x,y)\;\mathrm{d}x\mathrm{d} y = \displaystyle\int\!\!\displaystyle\int g(r,\theta)\;rd r\mathrm{d} \theta\) </p>
<p><img class="img-fluid" src="@@PLUGINFILE@@/Fig364_3.svg" alt="a graph showing the delta r in a segment"></p>
<p>We can see from the figure that the area elements change in size as \(r\) increases. The circumference of a circle of radius \(r\) is \(2\pi r\), so the length of an arc spanned by an angle \(\theta\) is \(2\pi r\dfrac{\theta}{2\pi}=r\theta\). Hence the area elements in polar coordinates are approximated by rectangles of width \(\delta r\) and length \(r\delta\theta\). Thus under the transformation from cartesian to polar coordinates we have the relation \[\displaystyle \delta x\delta y \to r\delta r \delta\theta\] that is, \(r\delta r \delta\theta\) plays the same role as \(\delta x\delta y\). This is why the \(r\) term appears in the integrand. Here \(r\) is playing the part of the Jacobian.</p>]]></text>
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