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.

Luisottifp 2022-09-25 Answered
Volume of torus
I 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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Answers (1)

Marvin Hughes
Answered 2022-09-26 Author has 6 answers
Step 1
V = π 2 2 ( 4 + ( 4 y 2 ) ) 2 ( 4 ( 4 y 2 ) ) 2 d y
Step 2
Outer 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
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