[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-207095-105":59,"doc-detail-207095-en":124},{"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":117,"head_meta":119,"extra_data":121,"updated_unix":123},105,"en","integration-math-problem-set","Integration - math problem set","","A collection of mathematics questions focused on calculus and algebraic modeling. Topics include solving intersections of curves, finding coordinates and equations of lines that are normals to curves, determining areas enclosed by curves and coordinate axes, and using integration to compute regions such as a leaf outline, a speed hump cross-section, and a river cross-section. Several problems also model motion using kinematic equations, requiring velocity, displacement, and distance calculations along with interpretation of changes over time.",{"@graph":69,"@context":116},[70,84,99],{"@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/integration-math-problem-set/207095/",{"url":83,"name":65,"@type":85,"author":86,"headline":65,"publisher":89,"fileFormat":92,"inLanguage":63,"description":67,"dateModified":93,"datePublished":93,"encodingFormat":92,"isAccessibleForFree":94,"interactionStatistic":95},"DigitalDocument",{"name":87,"@type":88},"Eden","Person",{"url":74,"name":90,"@type":91},"DocShare","Organization","application/pdf","2026-09-05",true,{"@type":96,"interactionType":97,"userInteractionCount":4},"InteractionCounter",{"@type":98},"ViewAction",{"@type":100,"mainEntity":101},"FAQPage",[102,108,112],{"name":103,"@type":104,"acceptedAnswer":105},"How do you find intersection points of two curves in these questions?","Question",{"text":106,"@type":107},"Set the two given equations equal and solve for x, then substitute back to get the corresponding y-coordinates.","Answer",{"name":109,"@type":104,"acceptedAnswer":110},"What method is used to find the area enclosed between curves and axes?",{"text":111,"@type":107},"Determine the region bounds from intersections, then integrate the difference between the upper and lower curve equations across the relevant x-interval.",{"name":113,"@type":104,"acceptedAnswer":114},"How are motion and distance calculated for the car and train problems?",{"text":115,"@type":107},"Use the given acceleration or velocity functions: integrate acceleration to obtain velocity, integrate velocity to obtain displacement, and apply limits for specific time intervals (e.g., first 20 seconds).","https://schema.org",{"og:url":83,"og:type":118,"og:title":65,"og:site_name":90,"og:description":67},"article",{"robots":120,"canonical":83},"index,follow",{"doc_id":122,"site_id":62},207095,1788597909,{"code":4,"msg":5,"data":125},{"doc_id":122,"user_id":126,"nickname":87,"user_avatar":127,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":128,"file_id":129,"file_url":130,"file_type":131,"file_size":132,"view_count":4,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":39,"language":133,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":134,"faqs":135,"seo_title":136,"seo_description":67,"update_tm":123,"read_time":46},1374391974468,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","## 11 The shape ABCD below represents a leaf.\n\nThe curve ABC has equation y=-x²+8x-9.  \nThe curve ADC has equation y=x²-6x+11.  \nx  \no  \n(i)Find algebraically the coordinates of A and C,the points where the curves intersect.  \n[5]  \n(ii)Find the area of the leaf.[7]  \n## 9 A car accelerates from rest.At time t seconds,its acceleration is given by a=4-0.2t ms-²untilt=20.\n\n(i)Find the velocity after 5 seconds.  \n[3]  \n(ii)What is happening to the velocity att=20?[1]  \n(iii)Find the distance travelled in the first 20 seconds.[3]  \n## 8 A train moves between two stations,taking 5 minutes for the journey.\n\nThe velocity of the train may be modelled by the equation v=60(t⁴-10t³+25t²)where v is measuredin metres per minute and t is measured in minutes.  \n## 11 The shaded region in the diagram shows a wooden shape.\n\nThe curve has equationand the coordinates of A are(-2,2).  \nx  \nThe line AB is the normal to the curve at the point A.  \n(i)Find the equation of the line AB.  \n[5]  \n(ii)Find the coordinates of the point B where the line AB meets the curve again.[3]  \n(iii)Find the shaded area.[4]  \n## 12 Fig.12 shows the shape AOB that is to be made from card.\n\nB is the point (5,0)and OB is part of the curve with equationy=0.3x²-1.5x.  \nThe line AB is the normal to the curve at B.  \nx  \nFig.12  \n④  \n(i)Find the equation of the line AB.The equation of the line AO is 2y+3x=0.  \n(ii)Find the coordinates of the point A.[3]  \n[5](iii)Find the area of the shape AOB.  \n## 9 The gradient function of a curve is given by\n\nFind the equation of the curve,given that it passes through the point (2,2).[4]  \n## 12 Two cars,A and B,move from rest away from a point Oon a straight road starting at the same time.\n\n(a)Car A moves with constant acceleration of 2ms-².Express the displacement of car A after time t seconds as a function of t.  \n口  \n(b)Car B moves with acceleration given by  \nExpress the displacement of car B after time t seconds as a function of t.[4]  \n(c)  (i)Find the time at which the cars are the same distance from O.  \n[2]  \n(ii)Find the distance they have travelled at that time.[2]  \n(d)Draw a sketch graph of the velocity of each car on the axes given.  \n[2]  \n## 14 The cross-section of a speed hump is modelled by the region enclosed by the x-axis and the curve\n\nThe graph is shown in Fig.14.Units are metres.  \n1.5  \n0.5  \nFig.14  \n(a)(i)Write down the maximum value of 1-(x-1)⁴.  \n[1]  \n(ii)Hence write down the maximum height of the speed hump.[1]  \n(b)Show that  \n[3]  \n[7]  \n(c)Find the area of the cross-section of the speed hump.  \n0 Sho ha  \nThe diagram shows part of the curvey=x²+2x-3.  \n(ii)Marc claims that the total area between the curve,the x-axis and the linesx=0 andExplain why he is wrong.[1]  \n(iii)Calculate the total area between the curve,the x-axis and the linesx=0 andx=2.  \n[3]  \n8 Amathematical gardener has a garden which is rectangular in shape measuring 20 metres by 8 metres.Hewishes to arrange the garden so that approximately halfof it is lawn and the rest flower bed.  \nHe sets up a coordinate system as shown in the diagram below and maps out the graph of the curve  \n0  \nShow that the area of the lawn is approximately 53%of the total area.  \n[6]  \n2 The gradient function of a curve that passes through the point(1,2)is given by  \n3 (i)Find the area enclosed between the curve y=8x³,the x-axis and the line x=2.  \n(ii)Hence,or otherwise,deduce the area between the x-axis,the y-axis,the line x=2 and the curvey=8x³+5.[1]  \n4  \n[4]  \n(ii)Interpret your answer geometrically.  \n[1]  \n11 Two curves,S₁and S₂have equations y=x²-4x+7 and y=6x-x²-1 respectively.The curves meet atA and at B.  \nFig.11  \n(i)Show that the coordinates of A and B are(1,4)and(4,7)respectively.  \n[2]  \nPoints PandQ lie on S₂and S₁between A and B.Pand Qhave the same x coordinate so that PQ is parallelto the y-axis,as shown in Fig.11.  \n(ii)Find an expression,in its simplest form,for the length PQas a function ofx.[2]  \n(iii)Use calculus to find the greatest lengt","cbCaif9bMSYwkGV3","https://ap.wps.com/l/cbCaif9bMSYwkGV3","pdf",1180399,"English","# Problem 8 - car acceleration and velocity\n# Problem 9 - gradient function and curve equation\n# Problem 11 - intersections of curves and area\n# Problem 12 - normal to a curve and area of a shape\n# Problem 14 - speed hump cross-section area\n# Motion problems - displacement and velocity graphs\n# River cross-section model - depth and volume","[{\"question\":\"How do you find intersection points of two curves in these questions?\",\"answer\":\"Set the two given equations equal and solve for x, then substitute back to get the corresponding y-coordinates.\"},{\"question\":\"What method is used to find the area enclosed between curves and axes?\",\"answer\":\"Determine the region bounds from intersections, then integrate the difference between the upper and lower curve equations across the relevant x-interval.\"},{\"question\":\"How are motion and distance calculated for the car and train problems?\",\"answer\":\"Use the given acceleration or velocity functions: integrate acceleration to obtain velocity, integrate velocity to obtain displacement, and apply limits for specific time intervals (e.g., first 20 seconds).\"}]","Integration - math problem set | PDF"]