[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-522925-105":3,"doc-detail-522925-en":81,"detail-sidebar-cat-0-en-105":97},{"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","volume-of-solids-with-known-cross-sections-volume-known-cross-sections","Volume of Solids with Known Cross Sections - Volume Known Cross Sections","","Worksheets for Infinite Calculus on finding the volume of solids formed from bases in the xy-plane and cross-sections perpendicular to an axis. Each problem defines a specific base region—semicircle, rectangle-constrained region, ellipse, or semicircle with given radius—and specifies the cross-section shapes and dimensions (squares, rectangles with height rules, or isosceles right triangles). Students compute volumes using the provided descriptions and may sketch the base from the given graph.",{"@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/volume-of-solids-with-known-cross-sections-volume-known-cross-sections/522925/",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/volume-of-solids-with-known-cross-sections-volume-known-cross-sections/522925.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},"What method do these worksheets use to find each solid's volume?","Question",{"text":63,"@type":64},"Each problem defines a base region and uses cross-sections perpendicular to an axis, specifying how cross-section dimensions depend on the geometry. The volume is then computed from the resulting cross-sectional areas.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"In Problem 1, what are the base and cross-sections?",{"text":68,"@type":64},"The base is the region enclosed by a semicircle (with its diameter on the x-axis) and the x-axis. Cross-sections perpendicular to the x-axis are squares.",{"name":70,"@type":61,"acceptedAnswer":71},"In Problem 3, what determines the cross-section shape and placement?",{"text":72,"@type":64},"The base is an ellipse with its major axis along the y-axis and a given minor axis. Cross-sections perpendicular to the x-axis are isosceles right triangles whose hypotenuse lies in the base.","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},522925,1791496582,{"code":4,"msg":82,"data":83},"success",{"doc_id":79,"user_id":84,"nickname":42,"user_avatar":85,"doc_module":4,"category_id":33,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":86,"file_id":87,"file_url":88,"file_type":89,"file_size":90,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":33,"language":91,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":92,"faqs":93,"seo_title":94,"seo_description":12,"update_tm":95,"read_time":96,"preview_page_count":4},962085570644,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","Kuta Software-Infinite Calculus  \nVolume of Solids with Known Cross Sections  \nName  Date  Period   \nFor each problem, find the volume of the specified solid. A graph representing the base is provided.  \n1) The base of a solid is the region enclosed by the semicircle y 􀀠 􀀕􀀘 􀀐 x 􀀕 and the x-axis. Cross-sections perpendicular to the x-axis are squares.  \n2) The base of a solid is the region enclosed by y 􀀠 􀀐 ~~ ~~x~~ ~~􀀕􀀜 􀀎 􀀗 and y 􀀠 􀀓 . Cross-sections perpendicular to the x-axis are rectangles with heights twice that of the side in the xy-plane.  \n3) The base of a solid is the region enclosed by an ellipse with a major axis of 􀀙 along the y-axis and a minor axis of 􀀗 . Cross-sections perpendicular to the x-axis are isosceles right triangles with the hypotenuse in the base.  \nFor each problem, find the volume of the specified solid. You may use the provided graph to sketch the base.  \n4) The base of a solid is the region enclosed by a semicircle with a radius of 􀀖, lying flat on the x-axis. Cross-sections perpendicular to the y-axis are squares.  \nKuta Software-Infinite Calculus  \nVolume of Solids with Known Cross Sections  \nName  Date  Period   \nFor each problem, find the volume of the specified solid. A graph representing the base is provided.  \n1) The base of a solid is the region enclosed by the semicircle y 􀀠 􀀕􀀘 􀀐 x 􀀕 and the x-axis. Cross-sections perpendicular to the x-axis are squares.  \n􀀘 ( 􀀕􀀘 􀀐 x 􀀕 ) 􀀕 dx  \n􀀐􀀘  \n= ~~ ~~􀀘~~ ~~􀀓􀀖􀀓~~ ~~ 􀁼 􀀔􀀙􀀙􀀑􀀙􀀙􀀚  \n2) The base of a solid is the region enclosed by y 􀀠 􀀐 ~~ ~~x~~ ~~􀀕􀀜 􀀎 􀀗 and y 􀀠 􀀓 . Cross-sections perpendicular to the x-axis are rectangles with heights twice that of the side in the xy-plane.  \n􀂳  \n3) The base of a solid is the region enclosed by an ellipse with a major axis of 􀀙 along the y-axis and a minor axis of 􀀗 . Cross-sections perpendicular to the x-axis are isosceles right triangles with the hypotenuse in the base.  \nFor each problem, find the volume of the specified solid. You may use the provided graph to sketch the base.  \n4) The base of a solid is the region enclosed by a semicircle with a radius of 􀀖, lying flat on the x-axis. Cross-sections perpendicular to the y-axis are squares.  \n􀀖 ( 􀀜 􀀐 y 􀀕 􀀎 􀀜 􀀐 y 􀀕 ) 􀀕 dy 􀀓  \n= 􀀚􀀕  \n􀂳  \nCreate your own worksheets like this one with Infinite Calculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCaiqHM9zMjxt2n","https://ap.wps.com/l/cbCaiqHM9zMjxt2n","pdf",79349,"English","# Volume of Solids with Known Cross Sections\n## Problem 1: Semicircle base, squares cross-sections\n## Problem 2: Region bounded by two curves, rectangles cross-sections\n## Problem 3: Ellipse base, isosceles right triangle cross-sections\n## Problem 4: Semicircle base with radius, squares cross-sections","[{\"question\":\"What method do these worksheets use to find each solid's volume?\",\"answer\":\"Each problem defines a base region and uses cross-sections perpendicular to an axis, specifying how cross-section dimensions depend on the geometry. The volume is then computed from the resulting cross-sectional areas.\"},{\"question\":\"In Problem 1, what are the base and cross-sections?\",\"answer\":\"The base is the region enclosed by a semicircle (with its diameter on the x-axis) and the x-axis. Cross-sections perpendicular to the x-axis are squares.\"},{\"question\":\"In Problem 3, what determines the cross-section shape and placement?\",\"answer\":\"The base is an ellipse with its major axis along the y-axis and a given minor axis. Cross-sections perpendicular to the x-axis are isosceles right triangles whose hypotenuse lies in the base.\"}]","Volume of Solids with Known Cross Sections - Volume Known Cross Sections | PDF",1791426313,10,{"code":4,"msg":82,"data":98},[99,103,107,110,115,120,124,129,134,137,140],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":100,"show_sort_weight":101,"slug":102},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":104,"show_sort_weight":105,"slug":106},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":108,"slug":109},70,"exam",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":112,"show_sort_weight":113,"slug":114},5,"Comic",60,"comic",{"id":116,"doc_module":4,"doc_module_name":25,"category_name":117,"show_sort_weight":118,"slug":119},6,"Technology",50,"technology",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":121,"show_sort_weight":122,"slug":123},"Healthcare",40,"healthcare",{"id":125,"doc_module":4,"doc_module_name":25,"category_name":126,"show_sort_weight":127,"slug":128},8,"Research & Report",30,"research-report",{"id":130,"doc_module":4,"doc_module_name":25,"category_name":131,"show_sort_weight":132,"slug":133},9,"Religion & Spirituality",20,"religion-spirituality",{"id":132,"doc_module":4,"doc_module_name":25,"category_name":135,"show_sort_weight":132,"slug":136},"World Cup","world-cup",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":138,"show_sort_weight":96,"slug":139},"Lifestyle","lifestyle",{"id":141,"doc_module":4,"doc_module_name":25,"category_name":142,"show_sort_weight":111,"slug":143},19,"General","general"]