Volume using shellsI'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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Volume using shellsI&#039;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&#039;t understand how to convert f(x) into an f(y). It doesn&#039;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?

Kyan Hamilton · Accepted Answer

Step 1Since the function describes a &quot;vertical parabola&quot;, integration in the  y-direction will involve using the &quot;right&quot; and &quot;left&quot; halves on this parabola, which will become the &quot;upper&quot; and &quot;lower&quot; halves when viewed from the y-axis. Solving for y gives us      x  =  1  ±      1    −    y       ..[The &quot;split&quot; occurs because the parabola represents a single function of  x, but not of  y.]Step 2The 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 &quot;lower arm&quot; of the parabola,      1     −         1    −    y         on the interval      0     ≤     y     ≤     1     ,, and the second, between the &quot;upper arm&quot; of the parabola,      1     +         1    −    y         and the lower arm on the interval      1     ≤     y     ≤     2     ..

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.

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