[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-456657-105":3,"doc-detail-456657-en":81,"detail-sidebar-cat-0-en-105":98},{"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","ap-calculus-bc-chapter-5-practice-test-definite-and-indefinite-integrals","AP Calculus BC Chapter 5 Practice Test - Definite and Indefinite Integrals","","AP Calculus BC Chapter 5 practice test focused on approximating areas under curves with left-endpoint rectangles and correcting estimates using exact integration. It includes right-hand Riemann sum approximations from tabulated function values, evaluation of multiple indefinite integrals (including trigonometric and substitution forms), and several definite integrals with piecewise definitions. A motion application uses a position function to compute displacement and distance traveled, followed by a full answer key.",{"@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/ap-calculus-bc-chapter-5-practice-test-definite-and-indefinite-integrals/456657/",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/ap-calculus-bc-chapter-5-practice-test-definite-and-indefinite-integrals/456657.png","ImageObject",300,407,{"name":42,"@type":43},"SANS","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-25","2026-09-30",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",19,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"How are areas under the curve approximated in this test?","Question",{"text":63,"@type":64},"Use 4 left endpoint rectangles for an over- or under-estimate, then compute the exact area using integration for comparison.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What method is used for the integral approximation based on a table?",{"text":68,"@type":64},"A right-hand Riemann sum is used, using the table values corresponding to the specified interval.",{"name":70,"@type":61,"acceptedAnswer":71},"How do you find the particle’s displacement and distance traveled?",{"text":72,"@type":64},"Displacement is the change in position over the interval, and distance traveled is the total path length based on the position function’s values over time.","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},456657,1790906326,{"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":91,"language":92,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":93,"faqs":94,"seo_title":95,"seo_description":12,"update_tm":96,"read_time":97},962090760505,"https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d","ID: 1  \nAP Calculus BC Name___________________________________  \nChapter 5 Practice Test Date________________ Period____  \n© 201 3 Ksuta S ofwtwa re LLMC . APll rights re servied .  \nFor each problem, approximate the area under the curve over the given interval using 4 left endpoint rectangles. For comparison, also find the exact area using integration. Explain why your answer is an over-or under-estimate.  \n1) y = −x + 3; [0, 2]  2  \n2) y = − ; [−4, −2]  \nx  \nFor each problem, use a right-hand Riemann sum to approximate the integral based off of the values in  \nthe table.  \n3) ∫019 f (x) dx  \n\n| x | 0 | 2 | 4 | 5 | 6 | 7 | 19 |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| f ( x) | 3 | 2 | 1 | 3 | 1 | 2 | 1 |\n\n4) ∫020 f (x) dx  \n\n| x | 0 | 1 | 3 | 13 | 15 | 18 | 20 |\n| --- | --- | --- | --- | --- | --- | --- | --- |\n| f ( x) | 7 | 8 | 7 | 5 | 3 | 2 | 0 |\n\nEvaluate each indefinite integral.  \n5) ∫ ~~ ~~15x4 dx 6) ∫ − ~~ ~~2544~~ ~~x dx  \n7) ∫ (−5x32 − 7x34~~ ~~ ) dx 8) ∫ ~~ ~~2(~~ ~~12x8~~ ~~83x~~ ~~6~~ ~~+~~ ~~1)~~ ~~ dx  \n© 2 01 3 KDu t a S o fPt war e LLWC . AlMlc riEght sR reserved. Mad e w irt hX InffdiJni5t e C ajl1culyu s .  \n-1-  \n9) ∫ ~~ ~~csinos2xx dx 10) ∫ ~~ ~~co10s2~~ ~~x dx  \n11) ∫ ~~ ~~7scoins2~~ ~~xx dx 12) ∫ ~~ ~~se1c~~ ~~x dx  \n13) ∫ 6sec 3x ⋅ tan 3xsec 3x dx 14) ∫ (4x3 + 1)4 ⋅ 36x2 dx  \n15) ∫ ~~ ~~(3x121)~~ ~~5~~ ~~ dx 16) ∫ ~~ ~~6scoins3~~ ~~3x3x dx  \n17) ∫ ~~ ~~sin~~ ~~2~~ ~~6(0x3x45~~ ~~+~~ ~~2) dx 18) ∫ ~~ ~~csc~~ ~~1(255xx5~~ ~~3)~~ ~~ dx  \n19) ∫ ~~ ~~20csc~~ ~~−5xco⋅~~ ~~cso2t(−5cscxsi5xn)~~ ~~(csc~~ ~~−5x) dx 20) ∫ ~~ ~~s4in⋅2se(tc2anxx) dx  \n© 2 0 13 KLu t a Softt wja r e LLC . AlLlK rijgh t sW reser ved.W MaEdje wikt h Intf9i3nGiut e C al2cAul4us .  \n-2-  \nEvaluate each definite integral.  \n21) −31 (−x +  −3x − 6  ) dx  \n23) 13  x 2 + x  dx  \n25) 14f (x) dx, f (x) = {xx  33,, xx  −22  \n27) 32−(x + 3)~~ ~~13 dx  \n29) ∫01~~ ~~(4x223)2~~ ~~ dx  \n© 2 0 1 3 Kujta S ofNt w ar e LjL Cp. ADlYlq riHg hxt sG re serv ed . MAaMd e wi0t h I nfzi9ni3te CaMlpcu luu s.  \nπ  \n−  \n22) ∫ 2 csc x ⋅ cot x dx  \n2π  \n−  \n3  \n24) 27f ( x) dx, f (x) = {x32, + 10x + 24, xx  −33  \n26) 21(x4 − x2 − x − 4) dx  \n28) 33f (x) dx, f (x) = {−2x2x− ,1,  \n30) ∫01−18x2 (3x3 − 1)2 dx  \nx ≤ 2  \nx > 2  \n-3-  \n31) ∫02 − ~~ ~~(x2~~ ~~2)~~ ~~2~~ ~~ dx 32) ∫014x(x2 − 1)2 dx  \nFor each problem, find F'(x).  \n33) F (x) = x14t dt 34) F (x) = 25x4(t + 3)~~ ~~12 dt  \n2 2  \n35) F (x) = ∫xx t~~ ~~12 dt 36) F (x) = ∫2x − ~~ ~~t13~~ ~~ dt  \n37) F (x) = ∫0x3 − ~~ ~~(t~~ ~~+42)2~~ ~~ dt 38) F (x) = ∫x2x(−t2 − 6t − 3) dt  \nA particle moves along a horizontal line. Its position function, in feet, is s(t) for t ≥ 0 seconds. For each problem, find the displacement of the particle and the distance traveled by the particle over the given interval.  \n39) s(t) = t3 − 28t2 + 196t; 1 ≤ t ≤ 9 40) s(t) = −t3 + 15t2 ; 9 ≤ t ≤ 15  \n© 2 0 1 3 Ku ta Soflt war e LLACH. AflJl0 ridgh t s reserved . Made wipthT I n fBiyn iYte Caldculgu sC.  \n-4-  \nAnswers to Chapter 5 Practice Test (ID: 1)  \n1) Exact: 4 2) Exact: −2 ln 2 + 2 ln 4 ≈ 1.386 3) 24 Approximate: ~~ ~~92 = 4.5 Approximate: ~~ ~~533420 ≈ 1.269  \n4) 84  5   5   5   7  \n5) − x 3 + C 6) −5x 4 + C 7) −2x 2 − 4x 4 + C  \n6 4 1 9) sec x + C 10) 10tan x + C 11) −7csc x + C  \n8) 4x + 4x − ~~ ~~ + Cx2  \n12) sin x + C 13) 32 ⋅ (sec 3x)43 + C 14) 35(4x3 + 1) 5 + C 15) − ~~ ~~2(3x21+~~ ~~1)4~~ ~~ + C  \n1 17) −4cot (3x5 + 2) + C 18) −5cos (5x5 − 3) + C  \n16) − ~~ ~~ + C sin 2 3x  \n19) 4sec (csc −5x) + C 20) cot (tan −4x) + C 21) 7 3 − 2 3  \n22)  3  ≈ −0.155  \n23) 173 ≈ 5.667 24) 493 ≈ 16.333 25) − ~~ ~~272 = −13.5 26) − ~~ ~~9910 = −9.9 3  \n27) −18~~ ~~4~~ ~~+~~ ~~3 ≈ −7.427 28) − ~~ ~~194 = −4.75 29) 47 ≈ 0.571  \n30) −6 2 2 1  \n31) −  3 ≈ −0.667 32) 3 ≈ 0.667 33) F ' (x) =  x  \n1   1   2   12x2   \n34) F ' (x) = 8(2x + 3) 2 35) F ' (x) = 2x2 − x 2 36) F ' (x) = − x 5 37) F ' (x) = − ( x3 + 2) 2  \n38) F ' (x) = −7x2 − 18x − 3 39) Displacement: 56  \nDistance traveled: ~~ ~~1131274 ≈ 419.037  \n40) Displacement: −486  \nDistan","cbCaietKbog1u5Lz","https://ap.wps.com/l/cbCaietKbog1u5Lz","pdf",52489,5,"English","# Chapter 5 Practice Test\n## Area under the curve with 4 left rectangles and exact integration\n## Riemann sum approximation using right endpoints\n## Evaluate indefinite integrals\n## Evaluate definite integrals\n## Find F'(x) from given F(x)\n## Particle motion: displacement and distance traveled\n# Answers to Chapter 5 Practice Test","[{\"question\":\"How are areas under the curve approximated in this test?\",\"answer\":\"Use 4 left endpoint rectangles for an over- or under-estimate, then compute the exact area using integration for comparison.\"},{\"question\":\"What method is used for the integral approximation based on a table?\",\"answer\":\"A right-hand Riemann sum is used, using the table values corresponding to the specified interval.\"},{\"question\":\"How do you find the particle’s displacement and distance traveled?\",\"answer\":\"Displacement is the change in position over the interval, and distance traveled is the total path length based on the position function’s values over time.\"}]","AP Calculus BC Chapter 5 Practice Test - Definite and Indefinite Integrals | PDF",1790746609,13,{"code":4,"msg":82,"data":99},[100,104,108,111,115,120,125,130,135,138,142],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":101,"show_sort_weight":102,"slug":103},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":105,"show_sort_weight":106,"slug":107},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":109,"slug":110},70,"exam",{"id":91,"doc_module":4,"doc_module_name":25,"category_name":112,"show_sort_weight":113,"slug":114},"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":121,"doc_module":4,"doc_module_name":25,"category_name":122,"show_sort_weight":123,"slug":124},7,"Healthcare",40,"healthcare",{"id":126,"doc_module":4,"doc_module_name":25,"category_name":127,"show_sort_weight":128,"slug":129},8,"Research & Report",30,"research-report",{"id":131,"doc_module":4,"doc_module_name":25,"category_name":132,"show_sort_weight":133,"slug":134},9,"Religion & Spirituality",20,"religion-spirituality",{"id":133,"doc_module":4,"doc_module_name":25,"category_name":136,"show_sort_weight":133,"slug":137},"World Cup","world-cup",{"id":139,"doc_module":4,"doc_module_name":25,"category_name":140,"show_sort_weight":139,"slug":141},10,"Lifestyle","lifestyle",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":143,"show_sort_weight":91,"slug":144},"General","general"]