Rotation creates solid circles like disks or washers like circles with holes
**NOTE: DIsk and washer are essentially the same, but with a disk the region we care about covers the entire interval over the axis of rotation, therefore making $r^2$ in the washer equation 0
| Disk | Washer | Shell | |
|---|---|---|---|
| When to use 1.1 | No gap between the region you are integrating and the axis of chosen/given rotation | When there IS a gap between the region you are integrating and the axis of chosen/given rotation | parallel slices to axis of rotation |
| Formula | Since it is a circle, integrating the area of a circle will therefore give you the volume. Area of a circle is πr^2 | It is just like the disk, but we are now accounting for this gap between the region and axis of rotation. Without rotating, then, we can expect a circle with a smaller hole in the center. This is why we subtract the hole (inner circle) from the entire circle (outer circle). Therefore, it’s similar to disk but we have to subtract the gap: π(R^2-r^2) where R is the outer radius and r is the inner radius (curve closest to axis of rotation) | distance from slice to axis (radius of the shell) times the height of the shell: V=2π ∫(radius)(height)d(variable) |
Radius r will just be the distance from axis of rotation
The height h, will just be the length of the slice or top-bottom or right-left | | When to use 1.2 | Slices are perpendicular to axis of rotation | Slices are perpendicular to axis of rotation | Slices are parallel to axis of rotation | | Example of 1.2 | Disk if axis of rotation is horizontal and you make a vertical slice with NO gap to respective axis of rotation
or
Disk if axis of rotation is vertical and you make a horizontal slice with NO gap to respective axis of rotation | Washer if axis of rotation is horizontal and you make a vertical slice (of course, if you also have that gap)
or
Washer if axis of rotation is vertical and you make a horizontal slice (+ the gap) | If axis of rotation is horizontal and you make a horizontal slice
or If axis of rotation is vertical and you make a vertical slice | | Mnemonics / Ways of remembering | dISK perpenDISCular (ahahahah) | PerpendicuLAR washER (they sound the same, well close enough ) | paraLLel sheLL
(think SAME=SHELL the s same helps lol)
also… SHELLS are ALONG the rotation so SHELLONG shlong hahaha |
Used when there is no hole, or when the slice touches the axis of rotation (no gap between region and axis)
$$ V=∫πr^2dx $$
where each slice is a tiny cylinder, and the volume of a cylinder is
$$ V=πr^2h $$
and the thickness is $h=dx$ therefore $dV=πr^2dx$. So to add all the slices we then get
$$ ∫πr^2dx $$
HOW TO FIND THE RADIUS:
Use when there is a hole
$$ V=∫π(R^2-r^2)dx $$
where R=outer radius and r=inner radius. Do this because you want to subtract the hole from the big circle.