1. Section 7.2 Day 1 Disk method
AP Calculus AB

2. Learning Targets – Day 1 Visualize a solid of revolution
Deconstruct and define each piece of the disk method formula
Calculate the volume of a solid of revolution about the x-axis
Calculate the volume of a solid of revolution about the y-axis

3. Visualize Solids of Revolutions
Find an Applet
Desmos?

4. Disk Method Formula X-Axis
Volume =𝑉= 𝜋 𝑎 𝑏 𝑅 𝑥 2 𝑑𝑥
Area of circle: 𝜋 𝑅 𝑥 2
𝑑𝑥=∆𝑥
Bounds 𝑥=𝑎 to 𝑥=𝑏

5. Disk Method Formula y-Axis
Volume =𝑉= 𝜋 𝑐 𝑑 𝑅 𝑦 2 𝑑𝑦
Area of Circle: 𝜋 𝑅 𝑦 2
𝑑𝑦= ∆𝑦
Bounds 𝑦=𝑐 to 𝑦=𝑑

6. Example 1: x-Axis
Find the volume of the solid formed by revolving the region bounded by the graph of 𝑦=−𝑥+1 from 𝑥=0 to 𝑥=1 about the x-axis.
𝜋 −𝑥 𝑑𝑥 = 1 3 𝜋

7. Example 2: X-axis
Find the volume of the solid formed by revolving the graph of 𝑦=4− 𝑥 2 from 𝑥=0 to 𝑥=2 about the x-axis.
𝜋 − 𝑥 𝑑𝑥 = 𝜋

8. Example 3: X-axis
Find the volume of the solid formed by revolving the region bounded by the graph of 𝑦= sin 𝑥 and the x-axis about the x-axis in the picture given.
𝜋 0 𝜋 sin 𝑥 2 𝑑𝑥 =2𝜋

9. Example 4: y-axis
Find the volume of the solid formed by revolving the region 𝑦= 𝑥 2 from 𝑥=0 to 𝑥=2 about the y-axis.
1. Change into 𝑥= 𝑦
2. 𝑦=0, 𝑦=4
3. 𝜋 𝑦 2 𝑑𝑦 =8𝜋

10. Example 5: y-axis
Find the volume of the solid formed by revolving the region 𝑦= 16− 𝑥 2 bounded by the x-axis & y-axis about the y-axis
1. 𝑥= 16− 𝑦 2
2. 𝑦=0, 𝑦=4
3. 𝜋 − 𝑦 𝑑𝑦 = 𝜋

11. Example 6: y-axis
Find the volume of the solid formed by revolving the region 𝑥=− 𝑦 2 +4𝑦 from 𝑦=1 to 𝑦=4 about the y-axis
𝜋 − 𝑦 2 +4𝑦 2 𝑑𝑦 = 𝜋