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Additional exercises guide the computation for piecewise-like function definitions over given interval endpoints.",{"@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/area-under-a-curve-using-limits-of-sums-evaluate-and-find-areas-under-curves/522902/",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/area-under-a-curve-using-limits-of-sums-evaluate-and-find-areas-under-curves/522902.png","ImageObject",300,407,{"name":42,"@type":43},"วิน","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-11-11","2026-10-08",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",9,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What does the worksheet ask you to evaluate first?","Question",{"text":63,"@type":64},"It asks you to evaluate each given sum in the “Evaluate each sum” section.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How are limits used in solving these problems?",{"text":68,"@type":64},"You evaluate each limit by analyzing the expression as n approaches infinity (lim n->∞), which arises from the sum structure.",{"name":70,"@type":61,"acceptedAnswer":71},"How do you find the area under the curve for the final problems?",{"text":72,"@type":64},"For each problem, you find the area under the curve over the given interval by setting up the solution using the limit as n goes to infinity of the upper sum.","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},522902,1791492331,{"code":4,"msg":82,"data":83},"success",[84,88,92,95,100,105,110,115,119,122,126],{"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":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":112,"show_sort_weight":113,"slug":114},8,"Research & Report",30,"research-report",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},"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},2336475104736,"https://ap-avatar.wpscdn.com/avatar/22000c4c5e0e5b17e70?x-image-process=image/resize,m_fixed,w_180,h_180&k=1786591360781797222","Kuta Software-Infinite Calculus  \nArea Under a Curve Using Limits of Sums  \nName  Date  Period   \nEvaluate each sum.  \n1) 􀀶n 􀀔􀀛 k 2) 􀀶n (􀀕k 􀀎 􀀘)  \nk 􀀠 􀀔 k 􀀠 􀀔  \n3) 􀀶n 􀀛 k􀀕 4) 􀀶n (k􀀕 􀀎 􀀗)  \nk 􀀠 􀀔 k 􀀠 􀀔  \nEvaluate each limit.  \n5)  kn 􀀔 (~~ ~~ 􀀎 k~~􀀕~~) 6)  kn 􀀔 (~~ ~~􀀘n 􀀎 kn􀀖~~􀀗 ~~)  \nFor each problem, find the area under the curve over the given interval. Set up your solution using the limit as n goes to 􀁦 of the upper sum.  \n7) f (x) 􀀠 􀀖 x 􀀎 􀀖 ; [􀀓 , 􀀖]  \n8) f (x) 􀀠 x 􀀕 􀀎 􀀘; [􀀔 , 􀀖]  \nKuta Software-Infinite Calculus  \nArea Under a Curve Using Limits of Sums  \nName  Date  Period   \nEvaluate each sum.  \nn  \n1) 􀀶 􀀔􀀛 k  \nk 􀀠 􀀔  \nn  \n􀀔􀀛 􀂘 􀀶 k  \n􀀔􀀛 􀂘 kn􀀠(􀀔n 􀀎 􀀔)􀀕  \n􀀜n 􀀕 􀀎 􀀜n  \nn  \n3) 􀀶 􀀛 k􀀕  \nk 􀀠 􀀔  \nn  \n􀀛 􀂘 􀀶 k􀀕  \n􀀛 􀂘 kn􀀠(􀀔n 􀀎 􀀔)(􀀕n 􀀎 􀀔)􀀙  \n~~ ~~􀀛~~ ~~􀀖~~ ~~ 􀀎 􀀗n 􀀕 􀀎 􀀗n􀀖  \n2) 􀀶n (􀀕k 􀀎 􀀘)  \nk 􀀠 􀀔  \nn n  \n􀀕 􀂘 􀀶 k 􀀎 􀀘 􀂘 􀀶 􀀔  \n􀀕 􀂘 kn􀀠(􀀔n􀀎~~ ~~􀀔􀀕~~ ~~) 􀀎k􀀔nn 􀀕 􀀎 􀀙n  \n4) 􀀶n (k􀀕 􀀎 􀀗)  \nk 􀀠 􀀔  \nn n  \n􀀶 k􀀕 􀀎 􀀗 􀂘 􀀶 􀀔  \nk 􀀠 􀀔 k 􀀠 􀀔  \nn(n 􀀎 􀀔)(􀀕n 􀀎 􀀔)􀀙  \n􀀖  \nn  \n􀀖  \n􀀎  \n􀀕  \nn  \n􀀕  \n􀀎  \n􀀕􀀘 n  \n􀀙  \n􀀎 􀀗n  \nEvaluate each limit.  \n5)  kn 􀀔 (~~ ~~ 􀀎 k~~􀀕~~)  \n (~~ ~~ 􀂘 kn 􀀔 􀀔 􀀎 􀀕n~~􀀕 ~~ 􀂘 kn 􀀔 k)  \n (~~ ~~ 􀂘 n 􀀎 􀀕n~~􀀕 ~~ 􀂘 n(n􀀎~~ ~~􀀔􀀕~~ ~~)~~ ~~)  \n (􀀖 􀀎 )􀀖  \n6)  kn 􀀔 (~~ ~~􀀘n 􀀎 kn􀀖~~􀀗 ~~)    \n􀀘  \n ( n􀀘n 􀂘􀂘 n􀀔􀀎 􀀎􀀗 􀂘􀀕 􀂘 􀂘k􀀶 k􀀎􀀖 ) 􀀕 )  \n (~~ ~~􀀕􀀔􀀗~~ ~~ 􀀎 􀀕􀀔n 􀀎 􀀗~~􀀕 ~~) ~~ ~~􀀕􀀔􀀗~~ ~~ 􀀠 􀀘􀀑􀀕􀀘  \nFor each problem, find the area under the curve over the given interval. Set up your solution using the limit as n goes to 􀁦 of the upper sum.  \n7) f (x) 􀀠 􀀖 x 􀀎 􀀖 ; [􀀓 , 􀀖]  \nkn 􀀔 (􀀖 (􀀓 􀀎 ~~ ~~􀀖~~ ~~~~ ~~􀀓 􀂘 k) 􀀎 􀀖 ) 􀂘 􀀖~~ ~~~~ ~~􀀓~~  ~~ kn 􀀔 (~~ ~~ 􀀎 􀀕􀀚n~~􀀕~~k~~ ~~ )  \n (~~ ~~ 􀂘 kn 􀀔 􀀔 􀀎 􀀚~~􀀕~~ 􀂘 kn 􀀔 k)  \n (~~ ~~ 􀂘 n 􀀎 􀀚~~􀀕~~ 􀂘 n(n􀀎~~ ~~􀀔􀀕~~ ~~)~~ ~~)  \n (~~ ~~􀀗􀀘􀀕 􀀎 􀀕􀀕􀀚n)  \n􀀗􀀘   \n􀀕  \n􀀠 􀀕􀀕􀀑􀀘  \n8) f (x) 􀀠 x 􀀕 􀀎 􀀘; [􀀔 , 􀀖]  \nlim  \nn 􀁯 􀁦  \nlim  \nn 􀁯 􀁦  \nn  \n􀀶  \nk 􀀠 􀀔  \nn  \n􀀶  \nk 􀀠 􀀔  \n(( 􀀔 􀀎 ~~ ~~􀀖~~ ~~~~ ~~􀀔~~ ~~ 􀂘 k) 􀀕 􀀎 􀀘 ) 􀂘 􀀖~~ ~~~~ ~~􀀔~~ ~~(~~ ~~ 􀀎 k~~􀀕~~ 􀀎 ~~ ~~􀀛kn~~􀀖~~􀀕~~ ~~)  \n (~~ ~~ 􀂘 kn 􀀔 􀀔 􀀎 􀀛n~~􀀕 ~~ 􀂘 kn 􀀔 k 􀀎 􀀛n~~􀀖 ~~ 􀂘 kn 􀀔 k􀀕)  \n (~~ ~~ 􀂘 n 􀀎 􀀛n~~􀀕 ~~ 􀂘 n(n􀀎~~ ~~􀀔􀀕~~ ~~) 􀀎 􀀛n~~􀀖 ~~ 􀂘 n(n~~ ~~􀀎~~ ~~􀀔~~ ~~)􀀙(􀀕n~~ ~~􀀎~~ ~~􀀔)~~ ~~)  \n (~~ ~~􀀘􀀙􀀖 􀀎  􀀎 ~~ ~~􀀖~~ ~~~~􀀕 ~~)  \n 􀀘􀀙   \n􀀖  \n􀁼 􀀔􀀛􀀑􀀙􀀙􀀚  \nCreate your own worksheets like this one with Infinite Calculus. Free trial available [at KutaSoftware.com](at KutaSoftware.com)","cbCaio7jcDqGcO7o","https://ap.wps.com/l/cbCaio7jcDqGcO7o","pdf",35605,"English","# Evaluate each sum\n# Evaluate each limit\n# Find the area under the curve (set up using n to infinity)","[{\"question\":\"What does the worksheet ask you to evaluate first?\",\"answer\":\"It asks you to evaluate each given sum in the “Evaluate each sum” section.\"},{\"question\":\"How are limits used in solving these problems?\",\"answer\":\"You evaluate each limit by analyzing the expression as n approaches infinity (lim n-\\u003e∞), which arises from the sum structure.\"},{\"question\":\"How do you find the area under the curve for the final problems?\",\"answer\":\"For each problem, you find the area under the curve over the given interval by setting up the solution using the limit as n goes to infinity of the upper sum.\"}]","Area Under a Curve Using Limits of Sums - Evaluate and Find Areas Under Curves | PDF",1791426291]