Adding up the volume of all of these shells would result in an integral like this: $$\int 2 \pi r h \ dr.$$ Long story short, we want to imagine our 3-D figure is made up of several infinitely thin cylindrical shells.
![shell method calculus shell method calculus](https://i.stack.imgur.com/LqCCW.png)
I’m not going to go into as much detail to explain where this integral comes from as I did in the cylinder method lesson, but if the following integral confuses you I’d recommend checking that lesson out by clicking on the above link. Result of rotating the region about the line \(y=-1\). I also went ahead and shaded the bounded region gray to make it a little easier to see (this was not done in Desmos). You should end up with something like the graph below. Go ahead and start with graphing all of the functions described in the problem. Graph the 2-D functionsĪs I always say, I suggest starting any problem possible by drawing what is being described to you. Just as before I’ll use the same 4 step process as in the cylinder method lesson. Other than that there isn’t much else to add so let’s jump into an example! Example 1įind the area of the solid created by rotating the area bounded between \(y= (x-1)^3-3\), \(y=-x-2\), and \(y=-2\) about the line \(y=-1\). It might help you make more sense of what’s going on if your start there. I’m not going to go into quite as much detail here as I did in that lesson. Please e-mail any correspondence to Duane Kouba byĬlicking on the following address heartfelt "Thank you" goes to The MathJax Consortium and the online Desmos Grapher for making the construction of graphs and this webpage fun and easy.Before reading through this problem, I’d recommend checking out my lesson on finding volumes of rotation using the cylinder shell method. Your comments and suggestions are welcome. Use the Shell Method to find the Volume of the torus.Ĭlick HERE to see a detailed solution to problem 9.Ĭlick HERE to return to the original list of various types of calculus problems. Form a torus (donut) by revolving this circle about the the vertical line $ x=-b $.
![shell method calculus shell method calculus](https://i.pinimg.com/originals/00/a1/ba/00a1ba1d9111ff1198da7114ff43e336.jpg)
This is a crucial step in the Shell Method process and is very important ! Then the Volume of the Solid of Revolution will be The radius $r$ and height $h$ will be CAREFULLY MARKED in region $R$. After sketching or imagining the resulting Solid of Revolution about the $y$-axis, it will be our job to create the dimensions of this cylindrical shell- the radius $r$, measured from the $y$-axis, and the height $h$, taken parallel to the $y$-axis at $x$. The details of the Shell Method are posted below.Īfter reading the description of region $R$, we will SKETCH and CAREFULLY LABEL this region $R$.
![shell method calculus shell method calculus](https://2aih25gkk2pi65s8wfa8kzvi-wpengine.netdna-ssl.com/hs/files/2018/01/Shell_problem2_solution.png)
Imagine the solid composed of thin concentric "shells" or cylinders, somewhat like layers of an onion, with centers of the shells being the $y$-axis.
![shell method calculus shell method calculus](https://miro.medium.com/max/552/0*suGTqhw6JDYkJhwE.png)
We start with a region $R$ in the $xy$-plane, which we "spin" around the $y$-axis to create a Solid of Revolution. The following problems will use the Shell Method to find the Volume of a Solid of Revolution. Volume of a Solid of Revolution Using the Shell MethodĬOMPUTING THE VOLUME OF A SOLID OF REVOLUTION USING THE SHELL METHOD