How can I find the volume of the solid generated when the region enclosed by $y=0, x=0, x=1$ and $(1+e^{-2x})^{0.5}$ is rotated through $360^\circ$ about the x axis?
Volume of a rotated region?
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calculus
geometry
integration
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0That makes no difference with the method you'll use to find the volume. – 2011-03-10
2 Answers
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I'm not completely sure that I understand what you mean, but try this: $\pi\int_0^1 f(x)^2 \text{ d}x$ where $f(x)$ in this case is your function $(1+e^{-2x})^{\frac{1}{2}}$.
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0Brilliant, I get it know. Cheers. – 2011-03-10
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The volume of an object can be found by integrating the cross-sectional area through all cross-sections.
A cross-section of this solid, taken perpendicularly to the $x$-axis at a distance $x$ from the $y$-axis, is a disk with radius $(1+e^{-2x})^{0.5}$. Express the area of this disk as a function of $x$, and then integrate this function from $x=0$ to $x=1$.