Washer or Disk Method ?

[Jason76]

Find the volume:

\(\displaystyle x = 2 \sqrt{5y}, x = 0, y = 5\) about the y axis

I think the way to find out which to use, is to set the x equations equal to each other, and solve for y. If 0, comes out as an intersection, then use the washer method.

\(\displaystyle 2 \sqrt{5y} = 0\)

\(\displaystyle y = 0\), so use washer method ?? Also, we know know the limits of integration, and they're \(\displaystyle 0\) lower bound, and \(\displaystyle 5\) upper bound (given)

 

[HallsofIvy]

A "washer" is a disk with a hole in it (go down to your local hardware store and ask to look at some washers). Here, you are rotating around the y-axis and you go from that axis to the graph of \(\displaystyle x= 2\sqrt{5x}\). Do see a hole?

 

[Jason76]

Any graph using using the washer method, and revolving around "y = 0" or "x = 0", will have one point that is (0,0) That seems to be way to figure it out. But if it revolves around say, y = 5, or x = 2. then it's different.

 

[Jason76]

As you were saying, and I've now noticed, the key is simply observation. The ones that are "washer method" just don't touch the axis of rotation.

 

[markraz]

to make this simple on yourself ..........disk and washer are the same exact formula the exception being with disk...... the g(x) is just always zero since there is only one radius. the outer radius is always f(x)

washer f(x)^2 - g(x)^2
disk f(x)^2 - g(0)^2

so just essentially disregard g(x) when you only have one function curve... If your teacher isn't an azz you could just leave in the g(0)^2 and get full credit since technically then the axis of rotation is the inner function..

I had to argue with my teacher about this since mistakenly she took off points because of it...... finally I had to go out of my way to show her a proof I pulled out of an old calculus book and she had finally to submit after I schooled her on it.........

Also just remember the f(x) is the outer most curve/line, and is perpendicular to the axis of rotation.

One other cool thing about washer, is you always will know you have selected the correct f(x) and g(x) if it is a positive solution.
So if you get a negative solution, then just swap your functions... you can't lose....