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If the region bounded by the upper function $y_{upper}=f(x)$ and the lower funciton $y_{lower}=g(x)$, and the lines $x=a$ and $x=b$ is revolved about the $x$-axis, then its volume of revolution is given by:$$ \begin{align} \displaystyleV &= \int_{a}^{b}{\Big([f(x)]^2-[g(x)]^2\Big)}dx \\&= \int_{a}^{b}{\Big(y_{upper}^2-y_{lower}^2\Big)}dx\end{align} $$ Example 1 Find the volume of revolution generated by revolving the region between $y=x^2$ and […]
Volume of Revolution We can use integration to find volumes of revolution between $x=a$ and $x=b$.When the region enclosed by $y=f(x)$, the $x$-axis, and the vertical lines $x=a$ and $x=b$ is revolved through $2 \pi$ or $360^{\circ}$about the $x$-axis to generate a solid, the volume of the solid is given by:$$ \begin{align} \displaystyleV &= \lim_{h […]
Volumes by Cross Sections can be setting a unit volume, \( \delta V \) of a cross section. Then establish an integration for evaluating the volume of the entire solid. Worked Examples of Volumes by Cross Sections The base of a solid is formed by the area bounded by \( y = \cos x\) and […]
Let’s consider the problem of finding the volume of the solid obtained by rotating about the \(x\)-axis or parallel to \(x\)-axis of the region, where the core idea of cylindrical shells method for finding volumes. We get a cylinder if we slice perpendicular to the \(y\)-axis. But to compute the washer’s inner and outer radius, […]