Chapter 5: Applications of Integration
Section 5.3: Volume by Slicing
The animation in Figure 5.3.1 shows a cutting plane intersecting a solid.
Suppose the slice at x exposes a region Rx with area Ax.
Further, suppose Rx is "thickened" by dx to a slab; the volume of this slab would be Ax dx.
The volume of the solid would be the sum of the volumes of the slabs, that is
Figure 5.3.1 Solid segmented by slicing
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 α with the horizontal.
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 yz-plane.
By the method of slicing, obtain the volume of the solid whose base in the xy-plane is the region bounded by the x-axis, and the curves y=sinx and x=π/2, and whose cross sections parallel to the yz-plane are equilateral triangles.
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