[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-522923-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-522923-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","volumes-by-cylindrical-shells-volume-cylinders","Volumes by Cylindrical Shells - Volume Cylinders","","Practice worksheet for using the method of cylindrical shells to compute volumes of solids formed by revolving a planar region about the y-axis or a specified axis. Problems include revolutions of regions bounded by linear and quadratic curves, along with required integral setups. A critical thinking item compares cylindrical-shell and washer/disk methods, highlighting why one method can be simpler. Includes graph guidance to sketch curves and shade enclosed regions.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/exam/","Exam",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/volumes-by-cylindrical-shells-volume-cylinders/522923/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/volumes-by-cylindrical-shells-volume-cylinders/522923.png","ImageObject",300,407,{"name":42,"@type":43},"Melati","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-10","2026-10-08",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",7,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"How do you use the cylindrical shells method for these volume problems?","Question",{"text":63,"@type":64},"Set up the volume as an integral in the appropriate variable using shell radius and shell height, then revolve the region about the specified axis.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"Why is problem 2 easier with cylindrical shells than with washers?",{"text":68,"@type":64},"The cylindrical shell method needs only one integral, while the washer/disk method would require two integrals for the region.",{"name":70,"@type":61,"acceptedAnswer":71},"What role do the provided graphs play in solving the worksheet problems?",{"text":72,"@type":64},"The graphs help sketch the curves, identify the enclosed region, and determine the correct bounds for the integral when revolving about the axis.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},522923,1791753226,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,109,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":93,"slug":94},70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"Technology",50,"technology",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":106,"show_sort_weight":107,"slug":108},"Healthcare",40,"healthcare",{"id":110,"doc_module":4,"doc_module_name":25,"category_name":111,"show_sort_weight":112,"slug":113},8,"Research & Report",30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":96,"slug":129},19,"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":33,"language":139,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":12,"update_tm":143,"read_time":123,"preview_page_count":4},962085570644,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","Kuta Software-Infinite Calculus Name   \nVolumes by Cylindrical Shells Date  Period   \nFor each problem, use the method of cylindrical shells to find the volume of the solid that results when the region enclosed by the curves is revolved about the y-axis.  \n1) y = x + 4 2  \ny = x + 4  \n2) y = 7 y = xx = 0 x = 4  \nCritical thinking question:  \n3) Solve problem 2 using the method of washers. Why is this problem easier using cylindrical shells?  \nFor each problem, use the method of cylindrical shells to find the volume of the solid that results when the region enclosed by the curves is revolved about the y-axis. You may use the provided graph to sketch the curves and shade the enclosed region.  \n4) y = 2x  \n2  \ny = x  \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\nFor each problem, use the method of cylindrical shells to 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 = −x2 + 7 y = x2 + 5 Axis: x = 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\nKuta Software-Infinite Calculus Name   \nVolumes by Cylindrical Shells Date  Period   \nFor each problem, use the method of cylindrical shells to find the volume of the solid that results when the region enclosed by the curves is revolved about the y-axis.  \n1) y = x + 4 2  \ny = x + 4  \n2π∫01x( x + 4 − (x2 + 4)) dx  \n3 = π 10  \n2) y = 7 y = xx = 0 x = 4  \n2π∫04x(7 − x ) dx  \n432 = π 5  \nCritical thinking question:  \n3) Solve problem 2 using the method of washers. Why is this problem easier using cylindrical shells?  \nπ ∫02 ( y2 )2 dy + π ∫2742 dy = 4352~~ ~~ π The cylindrical shell method requires one integral, while the disk method requires two.  \nFor each problem, use the method of cylindrical shells to find the volume of the solid that results when the region enclosed by the curves is revolv","cbCaihPsDlCKa6ev","https://ap.wps.com/l/cbCaihPsDlCKa6ev","pdf",83255,"English","# Volumes by Cylindrical Shells\n## Problems 1-2 (Revolve about the y-axis)\n## Critical thinking: Shells vs Washers\n## Problems 3-4 (Graph sketching guidance)\n## Problem 5 (Revolve about a given axis)","[{\"question\":\"How do you use the cylindrical shells method for these volume problems?\",\"answer\":\"Set up the volume as an integral in the appropriate variable using shell radius and shell height, then revolve the region about the specified axis.\"},{\"question\":\"Why is problem 2 easier with cylindrical shells than with washers?\",\"answer\":\"The cylindrical shell method needs only one integral, while the washer/disk method would require two integrals for the region.\"},{\"question\":\"What role do the provided graphs play in solving the worksheet problems?\",\"answer\":\"The graphs help sketch the curves, identify the enclosed region, and determine the correct bounds for the integral when revolving about the axis.\"}]","Volumes by Cylindrical Shells - Volume Cylinders | PDF",1791426312]