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

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!

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

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

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

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

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

Example 4: horizontal washer rotated about y = 2 Volume

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

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

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..