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Use The Washer Method to find out the volume.

It may not be difficult to figure out the volume of a cylinder, but when it is a relatively irregular or variable width volume, the calculation is much more troublesome. The Washer Method is one of the solutions to this situation.

When we mastered how to find the area of a quadratic function by dividing it into segments, people improved this method and applied it to finding the volume of a three-dimensional. Use the Washer/Shell method to find the volume. What I want to show today is the "Washer."



Think of an area, which is obtained by the intersection of two functions, and then this area enters a three-dimensional space and rotates around the y axis to obtain a volume.




The red line in the figure is 



The green line is 


The entire graphic rotates around the y axis to get a "cup", which will make it difficult for us to calculate the volume of this "cup", so we divide it into a circle and a circle of cushions, and calculate each one The volume of the mat can be added together to get an approximate volume.




This is the data obtained after dividing it into 10 parts. What is interesting is that the volume seems to be symmetrical (the volume of the first few pieces is the same after the latter). I have calculated many times, so I am sure that it is not a wrong number, it should be some kind of coincidence.\par

Therefore, we can get a similar volume through The Washer Method, as shown in the figure\par





So, through this method we can calculate that the volume is approximately 0.56676



We can get the volume of the intersecting area by calculating the volume of the two parts and then subtracting them.

Since it is rotating around the y axis, our equation is





After calculation, mine score is 0.57256

This result is very close, and we only need to divide it more carefully to get a more precise answer.


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