Solving a Disk Method Problem: What Info Needed?

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In summary, the volume of a solid with this information is found by solving the equation of the line for $y$ and entering the result into a disk method calculator.
  • #1
JProgrammer
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So I am trying to find the volume of a solid with this information given to me:

𝑥=0
𝑦=0
𝑦=−2𝑥+2

However, when I go to enter this information into a disk method calculator, I don't have enough information to enter into the calculator, such as the lower function and limits.

My question is: what information would I use to find the volume using the disk method?

Thank you.
 
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  • #2
What is the axis of rotation?
 
  • #3
Let's consider the line given by:

\(\displaystyle \frac{x}{h}+\frac{y}{r}=1\) where \(\displaystyle 0<h,r\)

Observing that this is the two-intercept form of a line, we should recognize that the graph of the line (in the first quadrant) will be as follows:

View attachment 5987

Now, if we take the $x$-axis as the axis of rotation, we should see that the solid of revolution will be a cone of radius $r$ and height $h$, and thus the volume $V$ of the solid of revolution will be (using the formula for the volume of a cone):

\(\displaystyle V=\frac{1}{3}\pi r^2h\)

So, we know what our goal is. Now, if we are going to use the disk method, then we can look at an element of the volume, which is naturally, a disk (actually a cylinder with height $H=dx$):

\(\displaystyle dV=\pi R^2H\)

Now, as mentioned, we have $H=dx$ and we observe that $R=y$, but we want $R$ in terms of $x$, hence we solve the equation of the line for $y$ to get $y$ as a function of $x$:

\(\displaystyle y=\frac{r}{h}(h-x)\)

And so we may now state:

\(\displaystyle dV=\pi\left(\frac{r}{h}(h-x)\right)^2\,dx=\frac{\pi r^2}{h^2}(x-h)^2\,dx\)

Hence:

\(\displaystyle V=\frac{\pi r^2}{h^2}\int_0^h(x-h)^2\,dx\)

Let:

\(\displaystyle u=x-h\implies du=dx\)

And we have:

\(\displaystyle V=\frac{\pi r^2}{h^2}\int_{-h}^0 u^2\,du=\frac{\pi r^2}{h^2}\cdot\frac{h^3}{3}=\frac{1}{3}\pi r^2h\)
 

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  • #4
Now, suppose we wish to check our work using the shell method...

\(\displaystyle 2V=2\pi RH\,dy=2\pi yx\,dy=2\pi y\cdot\frac{h}{r}(r-y)\,dy=\frac{2\pi h}{r}y(r-y)\,dy\)

Hence:

\(\displaystyle V=\frac{2\pi h}{r}\int_0^r y(r-y)\,dy=\frac{2\pi h}{r}\left[\frac{r}{2}y^2-\frac{1}{3}y^3\right]_0^r=\frac{\pi h}{3r}\left[3ry^2-2y^3\right]_0^r=\frac{\pi h}{3r}\left(3r^3-2r^3\right)=\frac{1}{3}\pi r^2h\quad\checkmark\)
 

FAQ: Solving a Disk Method Problem: What Info Needed?

What is the disk method and how is it used in problem solving?

The disk method is a mathematical technique used to find the volume of a solid when it is rotated around a specific axis. It is commonly used in calculus to solve problems involving solids of revolution.

What information is needed to solve a disk method problem?

To solve a disk method problem, you will need to know the function that defines the solid, the axis of rotation, and the limits of integration. Additionally, you may need to know the density of the solid if you are calculating its mass.

How do you determine the limits of integration in a disk method problem?

The limits of integration are determined by the points where the function intersects with the axis of rotation. These points are also known as the bounds of the solid. In some cases, you may need to use multiple functions and integrate each section separately.

Can you use the disk method to find the volume of a solid that is not a perfect circle?

Yes, the disk method can be used to find the volume of any solid that can be rotated around an axis. This includes solids with irregular shapes, as long as they can be represented by a function.

Are there any limitations to using the disk method?

The disk method may not be suitable for solids with holes or cavities, as these can result in negative volume. Additionally, if the function defining the solid is not continuous, the disk method may not provide an accurate solution.

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