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<title>
Triples...in cylindrical and spherical coordinates [12.8]
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<h1>
Triple Integrals in cylindrical and spherical coordinates [12.8]
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<p class="show"><img src="0g/12.8.saucer.jpg" class="big">

<h3>Cylindrical coordinates</h3>
<p class="show"><img src="0g/12.7.cylindricaldV.jpg" class="rightalign">
The figure shown is very approximately a "box" with volume
$$dV=(r\,d\theta)(dr)(dz)=r\,dr\,d\theta\,dz.$$

<h3>Spherical Polar coordinates</h3>
<p class="show"><img src="0g/12.7.sphericalpolardV.jpg" class="rightalign">
The figure shown is very approximately a "box" with volume
$$dV=(\rho\,d\phi)(\rho\sin\phi\,d\theta)(d\rho)=\rho^2\sin\phi\,d\rho\,d\theta\,d\phi.$$
<br>
<img src="0g/12.7.sphericalpolardV3.jpg"><br>
 
<h3>Examples</h3>
<p class="show"><img src="0g/12.8.prob56.jpg" class="big">



<h3 id="mtfuji">Todo</h3>
<p class="todo">Problem 41.  <b>Total work (energy) needed to create Mt Fuji</b>
~Right circular cone with...
<ul class="todo"><li>Radius of bottom=62,000 ft ~ 19 km=19,000m.
<li>Height of 12,400ft=3780m.
<li>Density of 200 lbs/ft^3 = 3,200 kg/m^3.
<li>Force of gravity (weight)=$mg$, where $m$ is mass (kg) and $g\approx 10$ m/s^2.  If $h$ (height) has units of meters, then $mgh$ will have units of <i>Joules</i>.
</ul>

<p>It may be easiest to set up the integral <i>as if you were only interested in the volume of Mt Fuji</i> first, and then add the energy after that.
<p>Sketch:<br>
<img class="justme" src="0g/12.8.fuji.jpg"><br>

<p>A large coal power plant, or nuclear plant, has power output of 2 gigawatts=2,000 megawatts=$2\times10^{9}$Joules/second:  In a year of continuous operation it puts out
$6.3\times 10^{16}$J.  
<p class="todo">How many months / years would such a power plant need to operate to produce the energy content you calculated above?

<h4 class="handout">Image credits</h4>
<p class="handout"><a href="https://www.flickr.com/photos/earlg/223768314">Earl G</a>


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