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Volumes of Solids of Revolution: Disks and Washers

Learn how to calculate volumes of solids using disks and washers in calculus. Understand the formulas and applications with step-by-step examples for a better comprehension.

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Volumes of Solids of Revolution: Disks and Washers

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  1. Section 6.2 ∆ Volumes Solids of Revolution • Two types: • Disks • Washers

  2. Disks – the theory • revolving function around an axis creating a SOLID piece formal definition: let S be a solid that lies between x = a and x = b. If the cross-sectional area S in the plane Px, through x and the perpendicular to the x-axis, is A(x), where A is a continuous function, then the volume of S is: don’t forget: same rules apply, when switching x and y!

  3. Disks – the application • How do we use this? • we use circular cross-sections • A=πr2 • Use “c” as the equation for the line of revolution; on the axes, c=0 • So… • Horizontally • Vertical axis

  4. Volume Example 1: around the x-axis (horizontal)

  5. rotated about y = 1 Volume Example 2: horizontal line of revolution

  6. rotated about y-axis converting x to y: Volume Example 3: around the y-axis (vertical)

  7. Washers • similar idea, but with a twist: a hole • essentially, you do the same thing twice Outer function Inner function

  8. rotated about y = 2 Volume Example 4: horizontal washer

  9. rotated about x = 2 convert: Volume Example 5: vertical washer

  10. rotated about y = 3 Volume Example 6: complex integration sometimes: you just KNOW it’s going to be ugly… so: cheat! use your calculator!!

  11. Homework: Pg. 423 (1-11)all *(1-6) Set up only. *(7-10) Set up and integrate by hand. *(11) graph on graph paper 4 times and include new lines of revolution. Set up and integrate by hand..

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