[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-523573-105":59,"doc-detail-523573-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","worksheets-approximating-area-under-a-curve-infinite-precalculus","Worksheets - Approximating Area Under a Curve - Infinite Precalculus","","Worksheet for Infinite Precalculus practice focused on approximating the area under a curve over a specified interval using left-endpoint rectangles. Problems 1–4 ask for approximations with 4 or 5 left-endpoint rectangles, supported by graphs and sketches of rectangles. A critical thinking question compares results across problems to determine when the approximation is less than or greater than the actual area and considers how using more, smaller rectangles affects accuracy.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/worksheets-approximating-area-under-a-curve-infinite-precalculus/523573/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/worksheets-approximating-area-under-a-curve-infinite-precalculus/523573.png","ImageObject",300,407,{"name":92,"@type":93},"dabgoudw","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-11","2026-10-08",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",11,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What method do the problems use to approximate the area under the curve?","Question",{"text":113,"@type":114},"They use left-endpoint rectangles over the given interval to estimate the area under the curve.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How do the rectangle counts differ across problems 1–4?",{"text":118,"@type":114},"Problems 1–2 use 4 left-endpoint rectangles, while problems 3–4 use 5 left-endpoint rectangles.",{"name":120,"@type":111,"acceptedAnswer":121},"How should you judge whether the approximation is less than or greater than the actual area?",{"text":122,"@type":114},"The worksheet asks you to compare the rectangle-based estimate with the actual area, then classify each problem as less than or greater than the actual value.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},523573,1791556991,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":14,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":24,"preview_page_count":4},3985747860484,"https://ap-avatar.wpscdn.com/davatar_276721f389ce27ea32af1340a28f341c","# Kuta Software - Infinite PrecalculusApproximating Area under a Curve\n\nName   \nDate   Period   \nFor each problem, approximate the area under the curve over the given interval using 4 leftendpoint rectangles.  \n1) y 􀀠 x 􀀎 􀀗 ; [􀀐􀀕 , 􀀙]  \ny  \n􀀔􀀗  \n􀀔􀀕  \n􀀔􀀓  \n􀀛  \n􀀐􀀛 􀀐􀀙 􀀐􀀗 􀀐  \n2)  \nx 􀀕  \ny 􀀠 􀀐 􀀕   \n􀀎 􀀙 ; [􀀐􀀖 , 􀀔]  \ny  \n 􀀔􀀗   \n 􀀔􀀕   \n\n| 􀀔􀀓   |  |  |  |  |  |  |  |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| 􀀐􀀛   | 􀀐􀀙   | 􀀐􀀗   | 􀀐􀀕   |  |  |  |  |\n|  |  |  |  | 􀀕   | 􀀗   | 􀀙 􀀛   |  |\n\nx  \nFor each problem, approximate the area under the curve over the given interval using 5 leftendpoint rectangles. You may use the provided graph to sketch the curve and rectangles.  \n3) y 􀀠 x 􀀕 􀀎 􀀕x 􀀎 􀀖 ; [􀀐􀀗 , 􀀔]  \ny  \n 􀀔􀀗   \n 􀀔􀀕   \n 􀀔􀀓   \n 􀀛   \n 􀀙   \n 􀀗   \n 􀀕   \n􀀐􀀛 􀀐􀀙 􀀐􀀗 􀀐􀀕 􀀕 􀀗 􀀙 􀀛  \nx  \n4) y 􀀠 ; [􀀕 , 􀀚]  \ny  \n 􀀔􀀗   \n 􀀔􀀕   \n 􀀔􀀓   \n 􀀛   \n 􀀙   \n 􀀗   \n 􀀕   \n􀀐􀀛 􀀐􀀙 􀀐􀀗 􀀐􀀕 􀀕 􀀗 􀀙 􀀛  \nx  \n# Critical thinking question:\n\n5) Look back at problems 1-4. In which problems does the approximated area appear less than theactual area? In which problems does the approximated area appear greater than the actual area?Would using more and smaller rectangles improve our approximation or make it worse?  \nG V  \n# Kuta Software - Infinite PrecalculusApproximating Area under a Curve\n\nName   \nDate   Period   \nFor each problem, approximate the area under the curve over the given interval using 4 leftendpoint rectangles.  \n1) y 􀀠 x 􀀎 􀀗 ; [􀀐􀀕 , 􀀙]  \ny  \n􀀔􀀗  \n􀀔􀀕  \n􀀔􀀓  \n􀀛  \n􀀐􀀛 􀀐􀀙 􀀐  \n􀀗􀀓  \nFor each problem,endpoint rectangles  \n3) y 􀀠 x 􀀕  \n\n| |  |  |  |\n| --- | --- | --- | --- |\n|  | 􀀐   | 􀀛   | 􀀐   |\n\n􀀐  \n􀀕􀀘  \n2)  \nx 􀀕  \ny 􀀠 􀀐 􀀕   \n􀀎 􀀙 ; [􀀐􀀖 , 􀀔]  \ny  \n 􀀔􀀗   \n 􀀔􀀕   \n 􀀔􀀓    \n   \n   \n   \n   \n   \n   \n   \n   \n 􀀛 x  \nusing 5 leftrectangles.  \n| 􀀐􀀛   | 􀀐   | 􀀙   | 􀀐   | 􀀗   | 􀀐   | 􀀕   | 􀀕   | 􀀗   | 􀀙   | 􀀛   |\n| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |\n\nx  \n 􀀕􀀜 􀀗  \n􀀠 􀀚􀀑􀀕􀀘  \n# Critical thinking question:\n\n5) Look back at problems 1-4. In which problems does the approximated area appear less than theactual area? In which problems does the approximated area appear greater than the actual area?Would using more and smaller rectangles improve our approximation or make it worse?  \nThe approximated areas for problems 1 and 2 appear to be less than the actual. Problems  \n3 and 4 appear greater. More rectangles would improve our approximation.  \nCreate your own worksheets like this one with Infinite Precalculus. Free trial available at KutaSoftware.com","cbCaieO5tB689DTR","https://ap.wps.com/l/cbCaieO5tB689DTR","pdf",129845,"English","# Approximating area under a curve\n## Problems 1–2 (4 left-endpoint rectangles)\n## Problems 3–4 (5 left-endpoint rectangles)\n## Critical thinking question (compare approximations)","[{\"question\":\"What method do the problems use to approximate the area under the curve?\",\"answer\":\"They use left-endpoint rectangles over the given interval to estimate the area under the curve.\"},{\"question\":\"How do the rectangle counts differ across problems 1–4?\",\"answer\":\"Problems 1–2 use 4 left-endpoint rectangles, while problems 3–4 use 5 left-endpoint rectangles.\"},{\"question\":\"How should you judge whether the approximation is less than or greater than the actual area?\",\"answer\":\"The worksheet asks you to compare the rectangle-based estimate with the actual area, then classify each problem as less than or greater than the actual value.\"}]","Worksheets - Approximating Area Under a Curve - Infinite Precalculus | PDF",1791427039]