Volumes of Revolution Day 3 Cylinders Shells Sometimes

Volumes of Revolution Day 3 Cylinders (Shells)

Sometimes it isn’t easy to get a washer or disk; when you rotate a curve and slice perpendicular to axis of rotation it doesn’t work mathematically. Think about this: y = sinx

What would the washer look like? This is the washer from a side view – they stack dy so the integration would have to be in terms of y. The problem is R = sin-1 y AND r = sin-1 y

Washers won’t work when the two sides of the curve are the same function. So what to do? There is a shape made parallel to the axis of rotation. We call them Shells or Cynlinders.

This is the general picture: What is the volume of a cylinder? Lets see what it looks like in animation

What is the areas of each shell? In general the area of each shell would be You will determine both r and h. One will be in terms of x; the other in terms of y. THEN you will substitute one variable for the other and integrate.

Remember: These shells will “nest” parallel to the axis of rotation. So, the stacking will be along that axis; therefore you will integrate in terms of that variable. There will be examples when multiple methods work. Use what you prefer or recognize first. Tonight, however, try only shells

Examples Find the volumes of the figures created 1. , , rotated around the y axis. 2. 3. axis. , y = 0, x = 4, rotated around the y , y = 0, x = 4, rotated around the x axis. (AS cylinders)

Reminder: 1. Cylinders will be parallel to axis of rotation (new concept) 2. Cylinder nesting will tell you whether you integrate in terms of y (dy) or x (dx).