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Math · Advanced Math
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3. A sheil is a right circular cylinder with a circular that the radius of the cylinder is infinitesimally larger than Suppose the height of the shell is h, the radius of the hole is, shell is △r, so that the radius of the cylinder is r + Δ hole drilled through its ceater such of the bole the radius and the thickness of the dr (a) One way to compute the volume of this shell is to imagine cutting it open along the edge, perpendicular to the base, and flattening it out to obtain a rectangular slab of thickness Δ, as shown above, what is the volume of this rectangular slab? (b) There is a slight distortion in volume when we latten out the shell to form a slab in (a). Compute the actual volume of the shell by subtracting the volume of the right circulur cylinder minus the volume of the circular hole (also a tight circular cylinder). The difference in your answers in (a) and (b) should be r(r),h, when we sum up the shells and take the limit as Ar goes to 0 to get a definite integral, the sum of these extra terms r(Ar)h has limit 0, because (Ar)2 is the square of an infinitesimal, and hence it goes to 0 infinitely faster than Δ. This justifies using the formula for the volume of a sheil in (a) when setting up a definite integral for volume of revolution.
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