I know that this integral was done using Shell Method ;however ,I don't understand the highlighted part.Why is 1 subtracted from y?
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Zoey
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I think it should have been $\int_{\frac12}^1 2\pi y(x-1)dy = \int_{\frac12}^1 2\pi y(\frac1y-1)dy$ – Kay K. Sep 03 '17 at 22:35
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Why do we subtract 1? – Zoey Sep 03 '17 at 22:42
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Because the height of each shell is $x-1$. – Kay K. Sep 03 '17 at 22:43
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$\frac{1}{y}-1$ is the length along the $x$ axis from the curve to the left endpoint of the interval, which is the height of a shell in the range $1/2<y<1$. – Ian Sep 03 '17 at 22:44