Additional Resources
OpenStax Online Textbook
How to Read a Math Textbook
Additional Instructional Resources
Physical Applications
Objectives
- Determine the mass of a one-dimensional object from its lineardensity function.
- Determine the mass of a two-dimensional circular object from its radial density function.
- Calculate the work done by a variable force acting along a line.
- Calculate the work done in pumping a liquid from one height to another.
- Find the hydrostatic force against a submerged vertical plate.
Summary
To measure the work done by a varying force in moving an object, we divide the problem into pieces on which we can use the formula and then use a definite integral to sum the work done on each piece.
To find the total force exerted by water against a dam, we use the formula ,to measure the force exerted on a slice that lies at a fixed depth, and then use a definite integral to sum the forces across the appropriate range of depths.
Because work is computed as the product of force and distance (provided force is constant), and the force water exerts on a dam can be computed as the product of pressure and area (provided pressure is constant), prob-lems involving these concepts are similar to earlier problems we did using definite integrals to find distance (via “distance equals rate times time”) and mass (“mass equals density times volume”).