How to determine inner and outer radii for 'Washer Method'?

[emergentecon]

Main Question or Discussion Point

I'm curious if there is a heuristic or method to determine the inner and outer radii as used in the Washer Method, WITHOUT needing to graph anything?

 

[HallsofIvy]

Strictly speaking you never have to "graph" if you can determine which value is larger. If you problem is to determine the volume of the region between y= f(x) and y= g(x), rotated about the x-axis, then the "outer radius" is the larger of f and g for all g in an interval while the "inner radius" is the smaller. While it is worth checking if f(x)= g(x) in the interval (whether the two graphs cross so one gives the inner radius for one part of the interval and the other for another part), that seldom happens. As long as the two curves do not cross, the "washer method" is exactly the same as doing two "disk method" calculations, then subtracting the smaller from the larger.

 

[emergentecon]

Wow thank you so much! The answer was staring me in the face - but your explanation made it clear!
Apologies for the lax use of the word 'graph'.

 

[Mark44]

I'm curious if there is a heuristic or method to determine the inner and outer radii as used in the Washer Method, WITHOUT needing to graph anything?

Notwithstanding anything that HallsOfIvy said, it doesn't hurt to graph the two functions, and to graph the solid of revolution that is formed from them. Having a good mental image of the relationship between the two functions is usually very helpful, so why would you want to skip this step in understanding?

 

[emergentecon]

Your point is entirely valid - I agree 100%.
I am new to mathematics, going the self-taught route (sort of), but have an exam of sorts.
My problem is I have not developed a natural intuition yet for how functions 'graph' so to speak - so wanted to find a way to approach the problems, without the need to sketch the functions.

 

[Mark44]

Your point is entirely valid - I agree 100%.
I am new to mathematics, going the self-taught route (sort of), but have an exam of sorts.
My problem is I have not developed a natural intuition yet for how functions 'graph' so to speak - so wanted to find a way to approach the problems, without the need to sketch the functions.

That's not a productive approach, in my opinion (and based on having taught mathematics in college for 19 years). Precalculus courses spend a lot of time on the graphs of a variety of functions: linear, quadratic, higher-degree polynomial, rational, exponential and logarithmic, and trig. If you are studying calculus without that background knowledge, you are at a distinct disadvantage.

My understanding of how the brain functions is that one of the halves is used in working with symbols, as in solving equations, and so on, and the other half is used to understand information presented in an image. If you don't have experience being able to quickly graph a function, it's as if you're attempting to do the problem with one half of your brain tied behind your back. The problems you're attempting to solve here, of finding the volume of a solid of revolution, benefit greatly from having a sketch of (1) the functions involved, and (2) the solid of revolution iself. Without a sketch, it is often difficult to determine the limits of integration or to know when you need two integrals instead of one.

Since you are teaching yourself calculus, I would strongly advise learning the precalc material that is usually considered prerequisite to the study of calculus.

 

[emergentecon]

That's not a productive approach, in my opinion (and based on having taught mathematics in college for 19 years). Precalculus courses spend a lot of time on the graphs of a variety of functions: linear, quadratic, higher-degree polynomial, rational, exponential and logarithmic, and trig. If you are studying calculus without that background knowledge, you are at a distinct disadvantage.

My understanding of how the brain functions is that one of the halves is used in working with symbols, as in solving equations, and so on, and the other half is used to understand information presented in an image. If you don't have experience being able to quickly graph a function, it's as if you're attempting to do the problem with one half of your brain tied behind your back. The problems you're attempting to solve here, of finding the volume of a solid of revolution, benefit greatly from having a sketch of (1) the functions involved, and (2) the solid of revolution iself. Without a sketch, it is often difficult to determine the limits of integration or to know when you need two integrals instead of one.

Since you are teaching yourself calculus, I would strongly advise learning the precalc material that is usually considered prerequisite to the study of calculus.

With all due respect, whilst your advice is well intentioned and I do appreciate it, you entirely missed the point. Whilst it would be nice to have the luxury of time, to take the perfect approach to learning, this is not based in reality. At times, short-term goals take precedence. I'm optimising my objective function, given some very real constraints. I did not however say, that I would not return to the work, or that I do not intend to develop a better understanding, or intuitive approach, to sketching functions. Learning is not a one-off process, it is continual, and reinforcing.

 

[Mark44]

With all due respect, whilst your advice is well intentioned and I do appreciate it, you entirely missed the point. Whilst it would be nice to have the luxury of time, to take the perfect approach to learning, this is not based in reality.

Actually, my advice is based on reality. What I'm saying is that you'll have a much tougher time reaching even your short-term goals if you limit your solving techniques to those that don't use a graph.

At times, short-term goals take precedence. I'm optimising my objective function, given some very real constraints. I did not however say, that I would not return to the work, or that I do not intend to develop a better understanding, or intuitive approach, to sketching functions. Learning is not a one-off process, it is continual, and reinforcing.

I agree, but reinforcement implies the reuse of a technique that you have some familiarity with. From what I gather from what you've said, there are some areas that you have no such familiarity.