**Application of Integrals**

Area + Volume + Work

A complete guide for solving problems involving area, volume, work and Hooke’s Law.

## Area Between Two Curves

43 min 4 Examples

- Overview of how to find area between two curves
- Example of finding area between curves given the limits of integration
- Example of finding area between curves where we have to find the limits of integration
- Example of finding area between curves where we must divide the region
- Example of finding area between curves using the y-axis
- Introduction of how area between curves helps us to find volume

## Volumes with Known Cross Sections

38 min 4 Examples

- Overview of how we find volume using area of a base and a cross-sectional slice
- All Formulas needed for finding volume using Cross Sections
- 4 Examples where we find volume using all of our cross sectional formulas, for areas that are both perpendicular to the x-axis and also the y-axis

## Solids of Revolution – Disk and Washer Method

59 min 7 Examples

- Overview of Solids of Revolution (Disk and Washer)
- Formula for both the Disk and Washer method
- Example of finding volume about the x-axis (disk)
- Example of finding volume about the y-axis (washer)
- Example of finding volume about the line y = a
- Example of finding volume about the line y = -a
- Example of finding volume about the line x = a
- Example of finding volume about the line x = -a
- Example of finding Area between curves, Volume using Cross Sections, Volume using Solids of Revolution, and finding perimeter of a shaded region.

## Solids of Revolution – Shell Method

18 min 3 Examples

- Overview of Cylindrical Shell Method
- Example #1 using the shell method perpendicular to the y-axis
- Example #2 using the shell method perpendicular to the y-axis
- Example #3 using the shell method perpendicular to the x-axis

## Work and Hooke’s Law

59 min 7 Examples

- Overview of Work and Hooke’s Law
- Hooke’s Law Example #1 of finding the work done in stretching a spring beyond its natural length
- Hooke’s Law Example #2 of finding the work done in stretching a spring beyond its natural length
- Hooke’s Law Example #3 of finding the force needed in stretching a spring beyond its natural length
- Review of Work and Force and Units
- Example #1 finding work needed to lift a bucket and rope
- Example #2 finding work needed to lift a coiled chain to full extension
- Example #3 finding work needed to lift a leaky bucket
- Example #4 finding the work needed to pump water out of a cylindrical tank

## Moments and Center of Mass

34 min 2 Examples

- Overview of what is the Center of Mass (centroid) and Moments
- Example #1 of finding the center of mass using integration by parts
- Example #2 of finding the center of mass using Integration by Parts and Half-Angle Identity