Volume of Disks & Washers Section 7.2 Calculus AP/Dual, Revised ©2017 viet.dang@humbleisd.net 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Review The oil drum has a height of 5 feet and a radius of 2 feet. Find the volume. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Calculus in Motion Link – Calc in Motion 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Animation 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Cutting Up a Lime RADIUS HEIGHT 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

Graphs Rotate Around 𝒙–Axis or Any Horizontal Line 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

Graphs Rotate Around 𝒚–Axis or Any Vertical Line 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers YouTube Clip http://www.youtube.com/watch?v=3Y1dFIIHiLY 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Calculus in Motion http://www.youtube.com/watch?v=3oAjcLD34kc 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Steps for Disk Method Draw the function Identify the type of shape and what the graph is in relations to (respects to 𝒙 or 𝒚-axis) Identify and/or solve for the intersection points Solve for 𝑹. Vertical Strip Equation: 𝑽=𝝅 𝒂 𝒃 𝑹 𝒙 𝟐 𝒅𝒙 where it rotates around the 𝒙–axis Horizontal Strip Equation: 𝑽=𝝅 𝒄 𝒅 𝑹 𝒚 𝟐 𝒅𝒚 where it rotates around the 𝒚–axis Simplify, Solve, and LABEL accordingly. DO NOT SEPARATE THE EQUATIONS 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 1 Find the volume of a solid formed by revolving 𝒚= 𝒙 , the 𝒙-axis and the line of 𝒙=𝟓 about the 𝒙-axis. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 1 Find the volume of a solid formed by revolving 𝒚= 𝒙 , the 𝒙-axis and the line of 𝒙=𝟓 about the 𝒙-axis. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 2 Find the volume of the solid of revolution generated by revolving the region bounded by the graph of 𝒚= 𝒙 𝟑 , the 𝒚-axis, and the lines 𝒚=𝟏 and 𝒚=𝟖 about the 𝒚-axis. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 2 Find the volume of the solid of revolution generated by revolving the region bounded by the graph of 𝒚= 𝒙 𝟑 , the 𝒚-axis, and the lines 𝒚=𝟏 and 𝒚=𝟖 about the 𝒚-axis. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Your Turn Find the volume of the solid of revolution generated by revolving the region bounded by the graph of 𝒚= 𝟗− 𝒙 𝟐 in the first quadrant about the 𝒙-axis. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Your Turn Find the volume of the solid of revolution generated by revolving the region bounded by the graph of 𝒚= 𝟗− 𝒙 𝟐 in the first quadrant about the 𝒙-axis. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Let’s Have Some Fun 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 3 Rotate the region enclosed by 𝒚= (𝐬𝐢𝐧 𝒙) 𝟏/𝟐 on the interval 𝟎≤𝒙≤𝝅 about the 𝒙–axis. Determine the volume of the solid formed. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 4 Rotate the region enclosed by 𝒚= (𝐬𝐢𝐧 𝒙) 𝟏/𝟐 on the interval 𝟎≤𝒙≤𝝅 about the 𝒙–axis. Determine the volume of the solid formed. When this shape is rotated, what does it look like? 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

Slice the Item Rotating Around to 𝒙–Axis 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 4 Rotate the region enclosed by 𝒚= 𝒙 𝟐 on the interval 𝒙=𝟎 and 𝒙=𝟐 about the 𝒙–axis. Determine the volume of the solid formed. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 4 Rotate the region enclosed by 𝒚= 𝒙 𝟐 on the interval 𝒙=𝟎 and 𝒙=𝟐 about the 𝒙–axis. Determine the volume of the solid formed. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 4 Rotate the region enclosed by 𝒚= 𝒙 𝟐 on the interval 𝒙=𝟎 and 𝒙=𝟐 about the 𝒙–axis. Determine the volume of the solid formed. What does it look like? 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 4 Rotate the region enclosed by 𝒚= 𝒙 𝟐 on the interval 𝒙=𝟎 and 𝒙=𝟐 about the 𝒙–axis. Determine the volume of the solid formed. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Determine 𝒅𝒙 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Your Turn Find the volume of a solid of revolution generated by revolving the region bounded by the graph of 𝒚= 𝒙 𝟑 , the 𝒙–axis, and the lines of 𝒙=−𝟏 and 𝒙=𝟐 about the 𝒙–axis 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 5 Rotate the region enclosed by 𝒚= 𝟏 𝟐 𝒙−𝟏 on the interval 𝒚=𝟎, 𝒚=𝟐 and 𝒙=𝟎 about the 𝒚–axis. Determine the volume of the solid formed. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 5 Rotate the region enclosed by 𝒚= 𝟏 𝟐 𝒙−𝟏 on the interval 𝒚=𝟎, 𝒚=𝟐 and 𝒙=𝟎 about the 𝒚–axis. Determine the volume of the solid formed. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 5 Rotate the region enclosed by 𝒚= 𝟏 𝟐 𝒙−𝟏 on the interval 𝒚=𝟎, 𝒚=𝟐 and 𝒙=𝟎 about the 𝒚–axis. Determine the volume of the solid formed. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 5 Rotate the region enclosed by 𝒚= 𝟏 𝟐 𝒙−𝟏 on the interval 𝒚=𝟎, 𝒚=𝟐 and 𝒙=𝟎 about the 𝒚–axis. Determine the volume of the solid formed. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

Slice it in Revolving the 𝒚–Axis Rotate the region enclosed by 𝒚= 𝟏 𝟐 𝒙−𝟏 on the interval 𝒚=𝟎, 𝒚=𝟐 and 𝒙=𝟎 about the 𝒚–axis. Determine the volume of the solid formed. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 6 Rotate the region enclosed by 𝒚=𝟒− 𝒙 𝟐 about the 𝒚–axis in the first quadrant. Determine the volume of the solid formed. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 6 Rotate the region enclosed by 𝒚=𝟒− 𝒙 𝟐 about the 𝒚–axis in the first quadrant. Determine the volume of the solid formed. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

Slice it in Rotating the 𝒚–Axis Rotate the region enclosed by 𝒚=𝟒− 𝒙 𝟐 about the 𝒚–axis in the first quadrant. Determine the volume of the solid formed. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Your Turn Rotate the region enclosed by 𝒚=𝟒− 𝒙 𝟐 about the 𝒙–axis in the first quadrant. Determine the volume of the solid formed. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Your Turn Rotate the region enclosed by 𝒚=𝟒− 𝒙 𝟐 about the 𝒙–axis in the first quadrant. Determine the volume of the solid formed. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

Slice it in Rotating the 𝒙–Axis Rotate the region enclosed by 𝒚=𝟒− 𝒙 𝟐 about the 𝒙–axis in the first quadrant. Determine the volume of the solid formed. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers The Washer Method Draw the function Identify the type of shape and what the graph is in relations to (respects to 𝒙 or 𝒚-axis) Identify and/or solve for the intersection points Solve for 𝑹 and 𝒓 Vertical Strip Equation: 𝑽=𝝅 𝒂 𝒃 𝑹(𝒙) 𝟐 − 𝒓(𝒙) 𝟐 𝒅𝒙 where it rotates around the 𝒙–axis Horizontal Strip Equation: 𝑽=𝝅 𝒄 𝒅 𝑹(𝒚) 𝟐 − 𝒓(𝒚) 𝟐 𝒅𝒚 where it rotates around the 𝒚–axis 𝑹 is the top/right radius and r is the bottom/left of the radius NOTE: Square the radius first. It is NOT 𝑹(𝒙) 𝟐 − 𝒓(𝒙) 𝟐 Simplify, Solve, and LABEL accordingly. DO NOT SEPARATE THE EQUATIONS 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 7 Find the volume 𝑽 of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚= 𝟐 𝒙 and 𝒚=𝟑−𝒙 about the 𝒙-axis. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 7 Find the volume 𝑽 of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚= 𝟐 𝒙 and 𝒚=𝟑−𝒙 about the 𝒙-axis. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 7 Find the volume 𝑽 of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚= 𝟐 𝒙 and 𝒚=𝟑−𝒙 about the 𝒙-axis. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 8 Find the volume 𝑽 of the solid of revolution generated by revolving the region bounded by 𝒚= −𝒙 𝟐 +𝟒, 𝒙=𝟏, and 𝒚=𝟎 about the 𝒚-axis. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 8 Find the volume 𝑽 of the solid of revolution generated by revolving the region bounded by 𝒚= −𝒙 𝟐 +𝟒, 𝒙=𝟏, and 𝒚=𝟎 about the 𝒚-axis. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 8 Find the volume 𝑽 of the solid of revolution generated by revolving the region bounded by 𝒚= −𝒙 𝟐 +𝟒, 𝒙=𝟏, and 𝒚=𝟎 about the 𝒚-axis. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Your Turn Find the volume 𝑽 of the solid of revolution generated by revolving the region bounded by 𝒚= −𝒙 𝟐 +𝟒, 𝒚=𝟐𝒙+𝟏, and 𝒙=𝟎 about the 𝒙-axis. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Your Turn Find the volume 𝑽 of the solid of revolution generated by revolving the region bounded by 𝒚= −𝒙 𝟐 +𝟒, 𝒚=𝟐𝒙+𝟏, and 𝒙=𝟎 about the 𝒙-axis. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 9 Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚=𝟐𝒙 and 𝒚= 𝒙 𝟐 about line 𝒚=−𝟓. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 9 Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚=𝟐𝒙 and 𝒚= 𝒙 𝟐 about line 𝒚=−𝟓. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 9 Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚=𝟐𝒙 and 𝒚= 𝒙 𝟐 about line 𝒚=−𝟓. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 9 Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚=𝟐𝒙 and 𝒚= 𝒙 𝟐 about line 𝒚=−𝟓. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 10 Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚=𝟐𝒙 and 𝒚= 𝒙 𝟐 about the line 𝒙=𝟐. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 10 Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚=𝟐𝒙 and 𝒚= 𝒙 𝟐 about the line 𝒙=𝟐. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 10 Find the volume of the solid of revolution generated by revolving the region bounded by the graphs of 𝒚=𝟐𝒙 and 𝒚= 𝒙 𝟐 about the line 𝒙=𝟐. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 11 Find the volume of a solid formed by revolving 𝒇 𝒙 = 𝒙 𝟐 +𝟐, 𝒈 𝒙 =𝟒− 𝒙 𝟐 about the horizontal line 𝒚=−𝟑. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 11 Find the volume of a solid formed by revolving 𝒇 𝒙 = 𝒙 𝟐 +𝟐, 𝒈 𝒙 =𝟒− 𝒙 𝟐 about the horizontal line 𝒚=−𝟑. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 11 Find the volume of a solid formed by revolving 𝒇 𝒙 = 𝒙 𝟐 +𝟐, 𝒈 𝒙 =𝟒− 𝒙 𝟐 about the horizontal line 𝒚=−𝟑. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Example 11 Find the volume of a solid formed by revolving 𝒇 𝒙 = 𝒙 𝟐 +𝟐, 𝒈 𝒙 =𝟒− 𝒙 𝟐 about the horizontal line 𝒚=−𝟑. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Your Turn Setup the volume of solid by rotating the region of 𝒇 𝒙 =𝟗− 𝒙 𝟐 and 𝒚=𝟏𝟐 for 𝟎,𝟑 about the line 𝒚=𝟏𝟓. Do not solve. 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

AP Multiple Choice Practice Question 1 (non-calculator) Region 𝑹 is bounded by 𝒇 𝒚 = 𝒚 𝟐 −𝟑 and 𝒈 𝒚 =𝟑𝒚+𝟏, determine the following expressions below that gives the volume when rotated about the line 𝒙=−𝟑. (A) 𝛑 −𝟐 𝟏𝟑 𝟔− 𝒚 𝟐 𝟐 − 𝟐−𝟑𝒚 𝟐 𝒅𝒚 (B) 𝛑 −𝟏 𝟒 𝟐−𝟑𝒚 𝟐 − 𝟔− 𝒚 𝟐 𝟐 𝒅𝒚 (C) 𝛑 −𝟏 𝟒 𝒚 𝟐 −𝟑 𝟐 − 𝟑𝒚+𝟏 𝟐 𝒅𝒚 (D) 𝛑 −𝟏 𝟒 𝟑𝒚+𝟒 𝟐 − 𝒚 𝟐 𝟐 𝒅𝒚 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

AP Multiple Choice Practice Question 1 (non-calculator) Region 𝑹 is bounded by 𝒇 𝒚 = 𝒚 𝟐 −𝟑 and 𝒈 𝒚 =𝟑𝒚+𝟏, determine the following expressions below that gives the volume when rotated about the line 𝒙=−𝟑. Vocabulary Connections and Process Answer and Justifications 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers

§7.2: Volume of Disks and Washers Assignment Worksheet 8/9/2019 6:23 AM §7.2: Volume of Disks and Washers