Section 7.2 Day 1 Disk method AP Calculus AB

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

Visualize Solids of Revolutions Find an Applet Desmos?

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

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

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 0 1 −𝑥+1 2 𝑑𝑥 = 1 3 𝜋

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 0 2 4− 𝑥 2 2 𝑑𝑥 = 256 15 𝜋

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

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 0 4 𝑦 2 𝑑𝑦 =8𝜋

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 0 4 16− 𝑦 2 2 𝑑𝑦 = 128 3 𝜋

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 1 4 − 𝑦 2 +4𝑦 2 𝑑𝑦 = 153 5 𝜋