This document discusses several applications of integrals including calculating displacement over time, total distance traveled, area between curves, and volume of solids of revolution using disc and washer methods. Specifically, it explains how to use integrals to find displacement given velocity over time, distance given velocity over time, and area between two curves by integrating the top curve minus the bottom curve between the boundaries. It also provides the formulas for finding the volume of a solid of revolution using the disc method which integrates the radius function squared or the washer method which integrates the outer radius squared minus the inner radius squared.

1. Application of Integrals

2. Displacement vs. Total Time Traveled Displacement= ∫ b V(t) dt a Distance Traveled= ∫ b V(t) dt a

3. Displacement vs. Total Time Traveled

4. Total Distance Traveled

5. Area Between Curves To find the area between two curves: Top – Bottom Right - Left *Make sure to use y boundaries and x=

6. Area Between Curves Examples: Determine the area of the region enclosed by and

7. Area Between Curves

8. Volume of a Solid of Revolution Disc Method: ∏ ∫ b (f(x))^2 dx; a Ex.

9. Volume of a Solid of Revolution Washer Method: V= ∏ ∫ b ((R)^2 -(r)^2) dx; a Around the x axis Y= dx (boundaries on the x axis) Around the y axis x= dy (boundaries on the y axis)

10. Example: