Application Of Definite Integral
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This document lists six topics that involve calculating volumes, lengths, centers of mass, areas of revolution, work, and fluid forces against walls using integration: 1) volumes, 2) lengths of plane curves, 3) centers of mass, 4) areas of surfaces of revolution, 5) work, and 6) fluid forces against planar walls.
Application Of Definite Integral
- 1. For finding … 1. Volumes, 2. Lengths of plane curves, 3. Centers of mass, 4. Areas of surfaces of revolution, 5. Work, and 6. Fluid forces against planar walls.
- 28. Summary In all of our volume examples, no matter how the cross-sectional area A(x) of a typical slab is determined, the definition of volume as the definite integral is the heart of the calculations we made.
- 29. • We defined the volume of a solid S as the definite integral where cross- sectional area A(x) is an integrable cross-sectional area of S. • The area A(x) was obtained by slicing through the solid with a plane perpendicular to the x-axis. • Now we slice through the solid using circular cylinders of increasing radii, like cookie cutters. • We slice straight down through the solid perpendicular to the x-axis, with the axis of the cylinder parallel to the y-axis.
- 30. Cookie Cutters • The vertical axis of each cylinder is the same line, but the radii of the cylinders increase with each slice. • In this way the solid S is sliced up into thin cylindrical shells of constant thickness that grow outward from their common axis, like circular tree rings. • Unrolling a cylindrical shell shows that its volume is approximately that of a rectangular slab with area A(x) and thickness This allows us to apply the same integral definition for volume as before.
- 70. Thank you … Thomas Calculus Chapter 5 70