y = 7ex, y = 7e−x, x = 1; about the y-axis

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jabjfoster jabjfoster

14-01-2017
Mathematics

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y = 7ex, y = 7e−x, x = 1; about the y-axis

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Anonymous Anonymous · Answer 1 · 2017-01-27T09:02:29-05:00

A "typical" cylindrical shell of wall thickness dx at x=x will have the length (in the y-direction) L = (e^x – e^-x).

Rotating this shell about the y-axis generates a volume dV = 2.π.x.L = 2.π.x.(e^x – e^-x).dx

Total volume V of all such cylindrical shells in the given range = 2.π.∫x.(e^x – e^-x).dx

= 4.π.[x.cosh(x) – sinh(x)] from 0 to 1. Now insert the limits and we obtain:

= 4.π.[1.543 – 1.175] = 4.624 cubic units