Finding radius for volume of revolutionI'm getting comfortable immediately recognizing the outer and inner radius when the axis of rotation is adjacent to the function or when my visible functions are strictly curves.Take for example the function f ( x ) = 8 x 3 bounded by y = 0 , x = 1 rotated about the line x = 2.Geometrically I think the outer radius R is the x distance between f(x) and x = 2. Likewise, the inner radius r is the x distance between the lines x = 1 and x = 2. However, with there being a gap and with constants involved, I'm having difficulty understanding how to determine inner and outer radius. What I did was say that R = f ( y ) + 1 , r = 1However, according to Symbolab, R = f ( y ) − 2 and r = 1 − 2.I don't understand how this works because it appears there is a negative radius, and then it appears the big radius doesn't go up to center of the hole of the solid. I thought the radius of a washer is from edge to center rather than outer edge to inner edge.I understand the purpose of everything else. I know volume is only interested in the actual object and not the empty space caused by the gap in our function, so I understand why we're subtracting the area of the little circle. I understand why we're summing up volume slice by slice via integration. I understand that dy means we're summing vertically. Etc. The one thing I don't understand is how radius is obtained the way it is.

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Finding radius for volume of revolutionI&#039;m getting comfortable immediately recognizing the outer and inner radius when the axis of rotation is adjacent to the function or when my visible functions are strictly curves.Take for example the function   f  (  x  )  =  8      x    3   bounded by   y  =  0  ,  x  =  1 rotated about the line   x  =  2.Geometrically I think the outer radius R is the x distance between f(x) and   x  =  2. Likewise, the inner radius r is the x distance between the lines   x  =  1 and   x  =  2. However, with there being a gap and with constants involved, I&#039;m having difficulty understanding how to determine inner and outer radius. What I did was say that  R  =  f  (  y  )  +  1  ,  r  =  1However, according to Symbolab,   R  =  f  (  y  )  −  2 and   r  =  1  −  2.I don&#039;t understand how this works because it appears there is a negative radius, and then it appears the big radius doesn&#039;t go up to center of the hole of the solid. I thought the radius of a washer is from edge to center rather than outer edge to inner edge.I understand the purpose of everything else. I know volume is only interested in the actual object and not the empty space caused by the gap in our function, so I understand why we&#039;re subtracting the area of the little circle. I understand why we&#039;re summing up volume slice by slice via integration. I understand that dy means we&#039;re summing vertically. Etc. The one thing I don&#039;t understand is how radius is obtained the way it is.

Jovan Mueller · Accepted Answer

Step 1Our region is   y  =  8      x    3    ,  y  =  0  ,  x  =  1 rotated around   x  =  2.I would solve this one with cylindrical shells, so inner radius and outer radius are not particularly relevant.  v  =  2  π      ∫    0    1    (  2  −  x  )  (  8      x    3    )     d  xStep 2If you really wanted to use washers you would say   x  =            y                        1          3                      2    ..The outer radius is   (  2  −  x  )  =  (  2  −            y              1        3              2    )The inner radius is 1.  V  =  π      ∫    0    8    (  2  −            y                        1          3                      2        )    2    −  1     d  yAnd, hopefully, these two integrals are equal.As far as   R  =  f  (  y  )  −  2  ,  r  =  1  −  2 from symbolab. It doesn&#039;t matter if these numbers are negative, you are going to be squaring them anyway.

Finding radius for volume of revolution. The function f(x)=8x^3 bounded by y=0, x=1 rotated about the line x=2.

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