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emergentecon
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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 said: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?
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.emergentecon said: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 said: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.
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.emergentecon said: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.
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.emergentecon said: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.
The inner and outer radii for the Washer Method can be determined by looking at the cross section of the shape being rotated around an axis. The inner radius is the distance from the center of rotation to the inner edge of the cross section, while the outer radius is the distance from the center of rotation to the outer edge of the cross section.
The inner and outer radii are necessary in order to accurately calculate the volume of a three-dimensional shape using the Washer Method. Without these measurements, the volume calculation cannot be completed.
For non-circular shapes, the inner and outer radii can still be measured by looking at the cross section of the shape. However, instead of measuring from the center of rotation to the edge, the radii will need to be measured at different points along the cross section.
No, the inner and outer radii must be different for the Washer Method to work. If the radii are the same, the shape being rotated would be a solid cylinder rather than a washer shape, and a different method would need to be used to calculate the volume.
The inner radius will always be smaller than the outer radius in the Washer Method. This is because the shape being rotated is hollow rather than solid, so the inner radius represents the empty space inside the shape while the outer radius represents the outer edge of the shape.