$V\={\int }_a^{b}A\left(x\right) \mathit{\ⅆ}x$ 

Example 5.3.1

By the method of slicing, obtain the volume of a wedge cut from a cylinder of radius $r$. In particular, let the axis of symmetry for the cylinder lie along the $z$-axis, the bottom face of the wedge in the plane $z\=0$, and the slanted face of the wedge in the plane that passes through the origin and that makes an angle $\mathrm{\α}$ with the horizontal.

Example 5.3.2

By the method of slicing, obtain the volume of the solid whose base is an equilateral triangle of side $s$, and whose plane sections are squares.  In particular, the equilateral triangle lies in the plane $z\=0$ and has a vertex at the origin, and an altitude along the $x$-axis. The square cross sections are parallel to the $\mathrm{yz}$-plane.

Example 5.3.3

By the method of slicing, obtain the volume of the solid whose base in the $\mathrm{xy}$-plane is the region bounded by the $x$-axis, and the curves $y\=\sin \left(x\right)$ and $x\=\mathrm{\π}\/2$, and whose cross sections parallel to the $\mathrm{yz}$-plane are equilateral triangles.