1. Section 7.2 Day 2 Disk Method
AP Calculus AB

2. Learning Targets
Decompose and define each piece of the disk method revolved around a line other than an axis
Calculate the volume of solids of revolutions about the line 𝑥=𝑎
Calculate the volume of solids of revolutions about the line 𝑦=𝑏

3. Disk Method Formula line 𝑦=𝑎
𝑉=𝜋 𝑎 𝑏 𝑅 𝑥 2 𝑑𝑥
1. Line 𝑦=𝑎: 𝑅 𝑥 −𝑎 is the new radius
2. Line 𝑦=−𝑎:𝑅 𝑥 − −𝑎
𝑜𝑟 𝑅 𝑥 +𝑎 is the new radius

4. Disk Method Formula line 𝑥=𝑏
𝑉=𝜋 𝑐 𝑑 𝑅 𝑦 2 𝑑𝑦
1. Line 𝑥=𝑏: 𝑅 𝑦 −𝑏 is the new radius
2. Line 𝑥=−𝑏:𝑅 𝑦 −(−𝑏)
or 𝑅 𝑦 +𝑏 is the new radius

5. Example 1: line 𝑦=𝑎
Find the volume of the solid formed by revolving the region bounded by 𝑓 𝑥 =2− 𝑥 2 and 𝑔 𝑥 =1 about the line 𝑦=1
1. R: 2− 𝑥 2 −1
2. π − − 𝑥 2 −1 2 𝑑𝑥 = 𝜋

6. Example 2: line 𝑦=𝑎
Find the volume of the solid formed by revolving the region bounded by 𝑦=4− 𝑥 and 𝑦=−2 about the line 𝑦=−2
1. R: 4− 𝑥
2. 𝜋 − − 𝑥 𝑑𝑥 =

7. Example 3: line 𝑦=𝑎
Find the volume of the solid formed by revolving the region bounded by 𝑦=− 𝑥 and 𝑦=3 about the line 𝑦=3
1. R: − 𝑥 2 +6−3
2. 𝜋 − − 𝑥 2 +6−3 2 𝑑𝑥 ≈52.237

8. Example 4: line 𝑥=𝑏
Find the volume of the solid formed by revolving the region bounded by 𝑦= 9− 𝑥 2 , the x-axis, and 𝑥=1 about the line 𝑥=1
1. R: 9− 𝑦 2 −1
2. 𝜋 − 𝑦 2 −1 2 𝑑𝑦 ≈21.472

9. Example 5 line 𝑥=𝑏
Find the volume of the solid formed by revolving the region bounded by 𝑥=2− 𝑦 2 and 𝑥=−2 about the line 𝑥=−2
1. R: 2− 𝑦 2 +2
2. 𝜋 − − 𝑦 𝑑𝑦 ≈

10. Example 6 line 𝑥=𝑏
Find the volume of the solid formed by revolving the region bounded by 𝑥=4− 𝑦 4 and 𝑥=−1 about the line 𝑥=−1
1. R: 4− 𝑦 4 +1
2. 𝜋 − − 𝑦 𝑑𝑦 ≈