Volume of torusI wanna calculate the volume of a torus that is created when the circlearea ( x − 4 ) 2 + y 2 ≤ 4 revolves around the y-axis. I don't understand what the difference would be if I calculated the volume of the circle ( x − 4 ) 2 + y 2 = 4 revolving around the y-axis. Doesn't the circlearea just create a solid torus instead of a hollow? Then comes the practical problem of finding a formula for the volume. My idea is that if the distance from the "right" and "left" side of the circle to origo is R and r respectively, then the area of the thin circles that constitute the torus is π ( R 2 − r 2 ). Then the height is dy and so we integrate from, if x denotes the radius of the circle, -r to r. Would this work?

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Volume of torusI wanna calculate the volume of a torus that is created when the circlearea   (  x  −  4      )    2    +      y    2    ≤  4 revolves around the y-axis. I don&#039;t understand what the difference would be if I calculated the volume of the circle   (  x  −  4      )    2    +      y    2    =  4 revolving around the y-axis. Doesn&#039;t the circlearea just create a solid torus instead of a hollow? Then comes the practical problem of finding a formula for the volume. My idea is that if the distance from the &quot;right&quot; and &quot;left&quot; side of the circle to origo is R and r respectively, then the area of the thin circles that constitute the torus is   π  (      R    2    −      r    2    ). Then the height is dy and so we integrate from, if x denotes the radius of the circle, -r to r. Would this work?

Marvin Hughes · Accepted Answer

Step 1  V  =  π      ∫          −      2              2                  (      4      +              (        4        −                  y          2                )            )        2    −            (      4      −              (        4        −                  y          2                )            )        2    d  yStep 2Outer radius   =  4  +      4    −          y      2      Inner radius   =  4  −      4    −          y      2        V  =      ∫          −      2              2            (    16    π          4      −              y        2              )    d  y      ∫          −      2              2            4    −          y      2        d  y  =  2  π  V  =  16  π  ∗  2  π  =  32      π    2

I wanna calculate the volume of a torus that is created when the circlearea (x-4)^2+y^2 leq 4 revolves around the y-axis.

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