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It also covers spherical coordinates, the second derivative test for critical points, and essential curve concepts such as arclength, unit tangent, curvature, and the Frenet frame.",{"@graph":14,"@context":72},[15,34,55],{"@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/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/presentations/","Presentations",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/final-exam-cheat-sheet-integral-and-calculus-review/573142/",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/final-exam-cheat-sheet-integral-and-calculus-review/573142.png","ImageObject",442,249,{"name":42,"@type":43},"mieayamfan","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":52,"interactionType":53,"userInteractionCount":33},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"What do line integrals require regarding orientation?","Question",{"text":62,"@type":63},"Line integrals depend on whether the vector field is paired with the path element in a way that tracks direction; the cheat sheet distinguishes cases where orientation matters versus not.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"Which theorems are included for converting between integral types?",{"text":67,"@type":63},"Green’s Theorem and Stokes’s Theorem are provided to relate boundary integrals and area/surface integrals under specified orientation conditions.",{"name":69,"@type":60,"acceptedAnswer":70},"How is the second derivative test applied on the cheat sheet?",{"text":71,"@type":63},"It uses the determinant D built from fxx, fyy, and fxy at a critical point and then classifies the point by the sign of D and fxx.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},573142,1792015511,{"code":4,"msg":81,"data":82},"success",[83,87,92,97,102,107,112,117,122],{"id":84,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":85,"slug":86},11,90,"presentations",{"id":88,"doc_module":22,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},12,"Resumes",80,"resumes",{"id":93,"doc_module":22,"doc_module_name":25,"category_name":94,"show_sort_weight":95,"slug":96},14,"Invoices",70,"invoices",{"id":98,"doc_module":22,"doc_module_name":25,"category_name":99,"show_sort_weight":100,"slug":101},15,"Posters",60,"posters",{"id":103,"doc_module":22,"doc_module_name":25,"category_name":104,"show_sort_weight":105,"slug":106},16,"Social Media",50,"social-media",{"id":108,"doc_module":22,"doc_module_name":25,"category_name":109,"show_sort_weight":110,"slug":111},17,"Forms",40,"forms",{"id":113,"doc_module":22,"doc_module_name":25,"category_name":114,"show_sort_weight":115,"slug":116},18,"Letters",30,"letters",{"id":118,"doc_module":22,"doc_module_name":25,"category_name":119,"show_sort_weight":120,"slug":121},21,"Paper Templates",5,"papers-templates",{"id":123,"doc_module":22,"doc_module_name":25,"category_name":124,"show_sort_weight":4,"slug":125},158,"General","general-158",{"code":4,"msg":81,"data":127},{"doc_id":78,"user_id":128,"nickname":42,"user_avatar":129,"doc_module":22,"category_id":84,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":33,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":30,"language":135,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":12,"update_tm":139,"read_time":22},962090893677,"https://ap-avatar.wpscdn.com/davatar_3d24733baf745e90a7e4bdd5f77d97b2","Final Exam Cheat Sheet  \nI do not promise that I will give you all of this cheat sheet on the 􀀌nal... I promise that I will give you any part of it that I believe that you need. (e.g. to make everything 􀀌t onto two pages, I will surely not give you all of the curve formulas.)  \nIntegral De􀀌nitions and Basic Formulas:  \nLine Integrals:  \nZC ~F(~r) 􀀏 d~r = Zab ~F(~r(t)) 􀀏 ~r0 (t) dt ; Orientation Matters! ZC f (~r) ds = Za b f (~r(t)) jj~r0 (t)jj dt ; Orientation Doesn't matter! ZC f (~r) dx = Za b f (x(t); y (t); z (t)) x0 (t) dt ; Orientation Matters!  \n~  \nSurface Integrals: With N := ~ru 􀀂 ~rv  0 ; and with ~r(R) = S we have ZZS ~F 􀀏 ~n dS = Z ZR ~F(~r(u; v)) 􀀏 ~N(u; v) du dv Orientation Matters! Z ZS f (~r) dS = Z ZR f (~r(u; v)) jj ~N(u; v)jj du dv Orientation Doesn't matter!  \nGreen's Theorem: If D is a region in the plane, and @D has positive orientation (i.e. has counter-clockwise orientation), then  \nZ@D P (x; y) dx + Q(x; y) dy = Z ZD 􀀒 @@Qx 􀀀 @@Py􀀓 dA :  \nStokes's Theorem:  \nZ ZS (r 􀀂 ~F) 􀀏 ~n dA = Z@S ~F 􀀏 d~r :  \n(If ~n is pointing right at you then orient @S in a positive fashion (i.e. counterclockwise fashion, typically) to make the identity hold.)  \nSpherical Coordinates:  \nx = 􀀚 sin 􀀞 cos 􀀒 y = 􀀚 sin 􀀞 sin 􀀒 z = 􀀚 cos 􀀞  \ndV = 􀀚2 sin 􀀞 d􀀚 d􀀞 d􀀒 :  \nSecond Derivative Test: Suppose the second partial derivatives of f are continuous on a disk with center (a; b) and suppose rf(a; b) = (0 ; 0) : Let  \nD = D(a; b) = fxx (a; b)fyy (a; b) 􀀀 fxy (a; b)2 :  \n(a) If D > 0 and fxx (a; b) > 0 ; then (a; b) is a local minimum.  \n(b) If D > 0 and fxx (a; b) \u003C 0 ; then (a; b) is a local maximum.  \n(c) If D \u003C 0 ; then (a; b) is a saddle point.  \nCurves:  \n~r(t) = (x(t); y (t); z (t))  \nArclength  \nUnit Tangent Vector  \nCurvature Vector  \nCurvature  \nPrinciple Unit Normal  \nBinormal  \ns (t) := Zat jj~r 0 (u)jj du (measuring from a to t)  \n~ ~v(t) ~r 0 (t)  \nT (t) := =  \njj~v(t) jj jj~r 0 (t) jj  \n~ ~ ~  \ndT   T 0 (t)   T 0 (t) = =  \nds jj~v(t) jj jj~r 0 (t) jj 􀀌 􀀌 d~T 􀀌 􀀌  \n􀀔 := 􀀌􀀌  􀀌􀀌  \n􀀌 􀀌 ds 􀀌 ~􀀌 ~  T 0 (t)  N (t) := 􀀌􀀌 ~ 􀀌􀀌  \n􀀌􀀌T 0 (t) 􀀌􀀌  \n􀀌􀀌 􀀌􀀌  \n~ ~ ~  \nB (t) := T(t) 􀀂 N (t)  \n~B(t) = jj~~ ~~~~ ~~00~~ ~~􀀂􀀂~~ ~~~~ ~~0000~~ ~~jj~a(t) = d2~~ ~~s ~dt2T + 􀀒 dsdt􀀓 2 􀀔 ~N = aT ~T + aN ~N  \naT = ~r 0 􀀏 ~r 00  aN = jj~r 0 􀀂 ~r 00 jj  􀀔 = jj~r 0 􀀂 ~r 00 jj  \njj~r 0 jj jj~r 0 jj jj~r 0 jj 3  \nCheat Sheet Bonus:  \ncos2 􀀒 = 1 + cos(2􀀒) sin2 􀀒 = 1 􀀀 cos(2􀀒) sin(2􀀒) = 2 sin 􀀒 cos 􀀒 :  \n2 ; 2 ;","cbCaioLIZrkKpXqi","https://ap.wps.com/l/cbCaioLIZrkKpXqi","pdf",96582,"English","# Integral Definitions and Basic Formulas\n## Line Integrals\n## Surface Integrals\n## Green's Theorem\n## Stokes's Theorem\n## Spherical Coordinates\n## Second Derivative Test\n## Curves\n## Cheat Sheet Bonus","[{\"question\":\"What do line integrals require regarding orientation?\",\"answer\":\"Line integrals depend on whether the vector field is paired with the path element in a way that tracks direction; the cheat sheet distinguishes cases where orientation matters versus not.\"},{\"question\":\"Which theorems are included for converting between integral types?\",\"answer\":\"Green’s Theorem and Stokes’s Theorem are provided to relate boundary integrals and area/surface integrals under specified orientation conditions.\"},{\"question\":\"How is the second derivative test applied on the cheat sheet?\",\"answer\":\"It uses the determinant D built from fxx, fyy, and fxy at a critical point and then classifies the point by the sign of D and fxx.\"}]","Final Exam Cheat Sheet - integral and calculus review | PDF",1791933174]