Description

Similar to applications of the derivative, we explore how integrals can be used. Sections in this topic include finding area and areas bounded by two graphs; volumes by slicing and by revolution; arc length; surfaces of reolution; average value of a function and the Mean Value Theorem for Integrals; rectilinear motion and the integral; work; liquid pressure and force; centers of mass; and the applications to Biology and Business.

Sections of this topic include:

1. Area and Area Bounded by Two Graphs

   We begin our investigation of the applications of integrals by finding areas under graphs as well as areas bounded by two graphs

2. Volumes by Slicing

   A useful application of the integral is finding volumes by finding areas of cross sections using slices.

3. Solids of Revolution.

   Imagine revolving a function around the x- or y- axis. Then using integrals, finding the resulting volume of this solid of revolution.

4. Arc Length

   Using a formula for arc length and the integral, we can find the length of a curve from one point to another.

5. Surfaces of Revolution

   Again, imagine revolving a function around the x- or y- axis. But instead of finding the volume of this solid, we find it's surface area.

6. Average Value of a Function and the Mean Value Theorem for Integrals

   Using the integral, we can find the average y value of a function over an interval.

7. Surfaces of Revolution

   Given the acceleration or velocity of a particle moving in a straight line, use the integral to find exactly its position function based on a time t.

8. Work (Coming Soon)

   In this section, we use integrals to find total work done in applications such as pumping fluids or lifting weights.

    

9. Liquid Pressure and Force (Coming Soon)

   Here, we use integrals to find liquid pressure and force on a plate in some fluid material.

10. Centers of Mass

    In this section, we explore an algebraic way to find the center of mass of a region.

11. Centers of Mass (Coming Soon)

    Using the integral, we can find the center of mass (called the centroid) of a surface.

12. Applications to Biology and Business (Coming Soon)

    We conclude this section with applying integrals to the subjects of biology (flow through an artery) and in business (consumer surplus).