Volumes for Two Functions

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} \displaystyle V &= \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 […]

Volumes using Integration

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} \displaystyle […]

Volumes by Cross Sections

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 […]

Volumes by Cylindrical Shells Method

Let’s consider the problem of finding the volume of the solid obtained by rotating about the $x$-axis or parallel to $x$-axis the region, where the core idea of cylindrical shells method for finding volumes. If we slice perpendicular to the $y$-axis, we get a cylinder. But to compute the inner radius and the outer radius […]