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Section 7.2 Volume by slicing
Objectives
-
Find volume by accumulating the areas of thin cross sections through a solid.
-
Select slices perpendicular to the
\(x\)- or
\(y\)-axis based on the geometry.
-
Model solids of revolution using disk and washer cross sections.
-
If a solid has cross sections perpendicular to the \(x\)-axis with area \(A(x)\text{,}\) then the volume of the solid over \(a\le x\le b\) is given by
\begin{equation*}
V=\int_{a}^{b}\dd{A}=\int_{a}^{b} A(x)\dd{x}\text{.}
\end{equation*}
If a solid has cross sections perpendicular to the \(y\)-axis with area \(A(y)\text{,}\) then the volume of the solid over \(c\le y\le d\) is given by
\begin{equation*}
V=\int_{c}^{d}\dd{A}=\int_{c}^{d} A(y)\dd{y}\text{.}
\end{equation*}
-
Disk Method for solids of revolution (about horizontal axis):
\begin{equation*}
V=\int_{a}^{b} \pii R(x)^{2}\dd{x}
\end{equation*}
-
Washer Method (about horizontal axis):
\begin{equation*}
V=\int_{a}^{b} \pii (R\tsub{outer}(x)^{2}-R\tsub{inner}(x)^{2})\dd{x}
\end{equation*}
