Find volume between two functions using the shell method. The functions given are f(x)=2x-x^2 and g(x)=x. It is reflected across the x-axis.

jlo2ni5x 2022-07-20 Answered
Volume using shells
I'm working on a problem of finding volume between two functions using the shell method. The functions given are f ( x ) = 2 x x ² and g ( x ) = x. It is reflected across the x-axis.
I solved this previously using washers but the problem asks to solve using two methods. I believe this is a dy problem, thus I am trying to convert the two functions into f(y) and g(y), but I don't understand how to convert f(x) into an f(y). It doesn't seem like it can be solved in terms of y. Maybe I am missing something after looking at this too long.
How would I solve this problem using shells? I am approaching this correctly?
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Answers (1)

Kyan Hamilton
Answered 2022-07-21 Author has 12 answers
Step 1
Since the function describes a "vertical parabola", integration in the y-direction will involve using the "right" and "left" halves on this parabola, which will become the "upper" and "lower" halves when viewed from the y-axis. Solving for y gives us   x = 1 ± 1 y   ..
[The "split" occurs because the parabola represents a single function of x, but not of y.]
Step 2
The parabola and the line meets at the origin and at (1,1). So integration along the y-axis is going to require two integrals, one between the line   x = y   and the "lower arm" of the parabola,   1     1 y     on the interval   0     y     1   ,, and the second, between the "upper arm" of the parabola,   1   +   1 y     and the lower arm on the interval   1     y     2   ..
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