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Includes multiple exercises with explicit equations, axis directions, and sample computed volumes, plus critical thinking tasks deriving the sphere-volume formula and analyzing a drill-bit cylindrical hole through a 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do you find the volume when a region is revolved about the x-axis?","Question",{"text":63,"@type":64},"Set up the volume integral using washers/disks: radii are determined by the top and bottom curves, then use the disk formula with an integral over the x-interval of the enclosed region.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What changes when revolving about a given (non-x) axis?",{"text":68,"@type":64},"Use the same enclosed region, but redefine the disk/washer cross-sections relative to the specified axis (often switching to integration with respect to y) to get correct radii.",{"name":70,"@type":61,"acceptedAnswer":71},"What is the purpose of the critical thinking questions?",{"text":72,"@type":64},"They ask for deeper derivations and applications, including deriving the sphere volume formula using disks and finding the volume after drilling a cylindrical hole through a 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problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the x-axis.  \nFor each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the given axis.  \nFor each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the given axis. You may use the provided graph to sketch the curves and shade the enclosed region.  \n5) y = x 2 − 2 y = −2 x = 2  \nAxis: y = −2  \n\n| | |  |  |  |  |  |  | y |  |  |  |  |  |  | |  |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n|  |  |  |  |  |  |  | 8 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | 6 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | 4 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | 2 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n| −8 | − | 6 | − | 4 | − | 2 |  |  | 2 |  | 4 |  | 6 |  |  | 8 x |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | −2 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | −4 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | −6 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  | −8\u003Cbr>|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n\n6) x = y + 3 x = ~~ ~~y2 + 3 Axis: x = 1  \n\n| | |  |  |  |  |  |  | y |  |  |  |  |  |  | |  |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n|  |  |  |  |  |  |  | 8 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | 6 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | 4 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | 2 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n| −8 | − | 6 | − | 4 | − | 2 |  |  | 2 |  | 4 |  | 6 |  |  | 8 x |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | −2 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | −4 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  | −6 |  |  |  |  |  |  |  |  |  |\n|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n|  | −8\u003Cbr>|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |\n\nCritical thinking questions:  \n7) Use the method of disks to derive the formula for the volume of a sphere of radius r.  \n8) A 6 cm diameter drill bit is used to drill a cylindrical hole through the middle of a sphere of radius 5 cm. What is the volume of the resulting object?  \nKuta Software-Infinite Calculus Name   \nVolumes of Revolution-Washers and Disks Date  Period   \nFor each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the x-axis.  \n1) y = −x2 + 1 y = 0  \nπ 11(−x2 + 1) 2 dx = ~~ ~~1615π ≈ 3.351  \n2) y = 2x + 2 y = x 2 + 2  \nπ ∫02 ((2x + 2) 2 − ( x2 + 2) 2 ) dx  \n48 = π ≈ 30.159  \n5  \nFor each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the given axis.  \n3) y = x + 1  \n2 y = x + 1  \nAxis: y = −1  \nπ ∫01 (( x + 2)2 − ( x2 + 2)2 ) dx = ~~ ~~4930π ≈ 5.131  \n4) x = − y2 + 2 x = y Axis: x = −2  \nπ 12((− y2 + 4) 2 − ( y + 2) 2 ) dy = ~~ ~~1058~~ ~~ π ≈ 67.858  \nFor each problem, find the volume of the solid that results when the region enclosed by the curves is revolved about the given axis. You may use the provided","cbCaimWkDqDRBfJF","https://ap.wps.com/l/cbCaimWkDqDRBfJF","pdf",79550,"English","# Volumes of Revolution - Washers and Disks - Volume\n## Revolution about the x-axis\n## Revolution about a given axis\n## Critical thinking questions","[{\"question\":\"How do you find the volume when a region is revolved about the x-axis?\",\"answer\":\"Set up the volume integral using washers/disks: radii are determined by the top and bottom curves, then use the disk formula with an integral over the x-interval of the enclosed region.\"},{\"question\":\"What changes when revolving about a given (non-x) axis?\",\"answer\":\"Use the same enclosed region, but redefine the disk/washer cross-sections relative to the specified axis (often switching to integration with respect to y) to get correct radii.\"},{\"question\":\"What is the purpose of the critical thinking questions?\",\"answer\":\"They ask for deeper derivations and applications, including deriving the sphere volume formula using disks and finding the volume after drilling a cylindrical hole through a sphere.\"}]","Volumes of Revolution - Washers and Disks - Volume | PDF",1791426314]