### Disk Method

**When rotating about the x-axis, our equation will be in terms of (x), or simply y = f(x).**

**V = π β« [R(x)2]dx**

**When rotating about the y-axis, our equation will be in terms of (y), or simply x = g(y).**

**V = π β« [R(y)2]dy**

**How to solve:**

**Identify the bounds.**

**f(x) is the function unless it is specified that there is the a shift.**

**If there is a shift, identify the area rotated and it will be plugged in a place of f(x).**

**Hint: The result will either be (f(x)+shift), (shift-f(x)), or (f(x)-shift).**

**Plug into the equation and solve the integral.**

### Washer Method

**Typo: The equation should be Ο instead of 2Ο.**

**When the solid is rotated about the y-axis, our equation will be in terms of (y). V = β« π ( R2outer- r2inner)dy When the solid is rotated about the x-axis, our equation will be in terms of (x). V = β« Ο (R2outer - r2inner)dx. Find the volume.**

**Example 1: Find the volume of the region generated by rotating Y= x2, and X = y2 about the x-axis.**

### Shell Method

**When rotating about the y-axis, our equation must be in terms of (x), so we would use this equation β V =2π β« (radius)(height of shell) dx**

**When rotating about the x-axis, our equation must be in terms of (y), so we would use this equation β V = 2π β« (radius)(height of shell) dy.**

**How to Solve:**

**1. Identify the bounds.**

**2. The equation should be in terms opposite to the axis that itβs being rotated about.**

**3.Β ****The radius will be x or y (depending on the axis of revolution) unless it is specified that itβs shifted. If there is a shift, identify the new radius. Hint: It will be (f(x)+shift),(f(x)-shift), or (shift-f(x)).**

**4. Height, is going to be the function.**

**5. Plug into equation and solve integral.**

Credits:

Created with images by skeeze - "scuba diver boat leaping" β’ tpsdave - "sea ocean water"