1 / 14

Lesson 7.2 : Volume - The Disk Method

Lesson 7.2 : Volume - The Disk Method. Objective: To find the volume of a solid with curved surfaces. First, we can start with our "representative rectangle" from area. However, we will be rotating this rectangle about the x-axis.

Download Presentation

Lesson 7.2 : Volume - The Disk Method

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Lesson 7.2: Volume - The Disk Method Objective: To find the volume of a solid with curved surfaces.

  2. First, we can start with our "representative rectangle" from area. However, we will be rotating this rectangle about the x-axis http://college.cengage.com/mathematics/blackboard/shared/content/learning_aids/rotgraphs/5906033.html We create a "disk" - which is a cylinder. We can find the area of a cylinder. V=πr2h To find the overall volume we will need calculus to find the volume of all the infinitely small, individual cylinders

  3. Q: So how do we find the volume of the whole region? http://college.cengage.com/mathematics/blackboard/shared/content/learning_aids/rotgraphs/5906034.html A: Creating an infinite number of cylinders that represent the entire solid.

  4. We could put any solid on a graph to get an equation

  5. Let's see the progression:

  6. So let's change our equation for the entire solid where the curved boundary of the solid is defined by a function R(x) V = π r2 h What represents the radius of the disk? What determines the height of the disk? ΔV = π (R(x))2 Δx Now, find the volume as a limit of infinitely many disks.

  7. Remember from area, we can calculate this rotating with respect to the x or y

  8. You need to be very careful about how you think about the radius! Problem Set 7.2.1

  9. The Washer Method Occurs when there is a "hole" in the solid The volume equation then becomes Let R(x) = outer radius, and r(x) = inner radius

  10. Using Two Integrals With Respect to Y

  11. Volume of Other Solids - With Known Cross Sections The method using cylinders can also apply the same method with other cross sections We can calculate volume as long as we have a formula for the area of the cross section Across-section = side2 So we can write two equations: Horizontal rotation: Vertical rotation:

More Related