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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.

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Lesson 7.2 : Volume - The Disk Method

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  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:

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