[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-461681-105":59,"doc-detail-461681-en":130},{"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":123,"head_meta":125,"extra_data":127,"updated_unix":129},105,"en","1802-practice-exam-3b-solutions","18.02 Practice Exam 3B - Solutions","","Practice Exam 3B solutions for a multivariable calculus course topic set. Covers changing the order of integration, computing mass of a half-annulus with density depending on x and y, constructing mass-center integrals and using symmetry to justify ̅x=0. Includes verifying a vector field is conservative, finding a potential, evaluating a line integral over a specified curve, and computing work around a unit circle using both parametrization and Green’s theorem. Also addresses work on a rectangular boundary via Green’s theorem and a volume problem solved with change of variables for two quadratic surfaces.",{"@graph":69,"@context":122},[70,84,105],{"@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/1802-practice-exam-3b-solutions/461681/",{"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/1802-practice-exam-3b-solutions/461681.png","ImageObject",300,407,{"name":92,"@type":93},"anakgang17","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-11-04","2026-09-30",true,{"@type":102,"interactionType":103,"userInteractionCount":81},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How do you exchange the order of integration for Problem 1?","Question",{"text":112,"@type":113},"You redraw the integration region for the original bounds and then restate the double integral in the dxdy order, which splits into two pieces as indicated by the warning.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"Why does the center of mass x-coordinate ̅x equal 0 in Problem 2b?",{"text":117,"@type":113},"The integrand xρ is odd with respect to the y-axis, while both the region and density are symmetric under (x,y) → (-x,y), causing the integral to cancel to zero.",{"name":119,"@type":110,"acceptedAnswer":120},"How is Green’s theorem used in Problem 4b?",{"text":121,"@type":113},"The line integral around the unit circle is converted to a double integral over the unit disk using Green’s theorem, replacing (N x - M y) with the corresponding expression in the integrand.","https://schema.org",{"og:url":83,"og:type":124,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":126,"canonical":83},"index,follow",{"doc_id":128,"site_id":62},461681,1790762150,{"code":4,"msg":5,"data":131},{"doc_id":128,"user_id":132,"nickname":92,"user_avatar":133,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":81,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":81,"language":139,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":67,"update_tm":129,"read_time":39,"preview_page_count":4},962090883568,"https://ap-avatar.wpscdn.com/davatar_a8503ba1806abce46bf441b54a3ca4cd","18.02 Practice Exam 3B  \nProblem 1. a) Draw a picture of the region of integration of 􀁚01 􀁚x2x dydx  \nb) Exchange the order of integration to express the integral in part (a) in terms of integration in the order dxdy. Warning: your answer will have two pieces.  \nProblem 2. a) Find the mass M of the upper half of the annulus,  \n y   \n1 \u003C x2 + y 2 \u003C 9 (y 􀀕 0) with density 􀀎 = x2 + y 2 .  \nb) Express the x-coordinate of the center of mass, x¯, as an iterated integral. (Write explicitly the integrand and limits of integration.) Without evaluating the integral, explain why ¯x = 0.  \nProblem 3. a) Show that F = (3x2 − 6y2 )ˆi + (−12xy + 4y)ˆj is conservative.  \nb) Find a potential function for F.  \nc) Let C be the curve x = 1 + y3 (1 − y)3 , 0 􀀔 y 􀀔 1. Calculate 􀁚C F · dr.  \nProblem 4. a) Express the work done by the force field F = (5x + 3y)ˆi + (1 + cosy)ˆjon a particle moving counterclockwise once around the unit circle centered at the origin in the form 􀁚ab f (t)dt. (Do not evaluate the integral; don’t even simplify f (t) .)  \nb) Evaluate the line integral using Green’s theorem.  \nProblem 5. Consider the rectangle R with vertices (0 , 0), (1 , 0), (1 , 4) and (0 , 4) . The boundary of R is the curve C , consisting of C1 , the segment from (0 , 0) to (1 , 0), C2 , the segment from (1 , 0) to (1 , 4), C3 the segment from (1 , 4) to (0 , 4) and C4 the segment from (0 , 4) to (0 , 0) . Consider the vector field  \nF = (cos x sin y)ˆi + (xy + sin x cosy)ˆj  \na) Find the work of F along the boundary C oriented in a counterclockwise direction.  \nb) Is the total work along C1 , C2 and C3 , more than, less than or equal to the work along C?  \nProblem 6. Find the volume of the region enclosed by the plane z = 4 and the surface  \nz = (2x − y)2 + (x + y − 1)2 .  \nSuggestion: change of variables.  \n18.02 Practice Exam 3B — Solutions  \n􀀋 (1 , 2)  \n􀁏􀁏 􀀋  \n􀀋  \n􀀋  \n􀀋  \n􀀋  \n􀀋  \n1. a) y=2x􀀋􀀋􀀋􀀋􀀋􀀋􀀋 x=1 b) 􀁚01 􀁚yy/2 dxdy + 􀁚12 􀁚y 1/2 dxdy.  \n􀀋􀀋􀀋􀀋 􀁰􀁰􀁰􀁰􀁰 (1, 1)  \n􀀋􀀋􀀋 􀁰􀁰􀁰􀁰yx  \n􀀋􀁰􀁰􀁰 􀀯  \n􀀯  \n2. a) 􀀎dA = r sin 􀀒 rdrd􀀒 = sin 􀀒drd􀀒 .  \nr2  \nM = 􀁚 􀁚R 􀀎dA = 􀁚0􀀙 􀁚13 sin 􀀒 drd􀀒 = 􀁚0 􀀙 2 sin 􀀒d􀀒 = 􀀂− 2 cos 􀀒 􀀃 􀀙0 = 4.  \nb) x¯ = 1M 􀁚 􀁚R x􀀎dA = 14 􀁚0􀀙 􀁚13 r cos 􀀒 sin 􀀒drd􀀒  \nThe reason why one knows that x¯ = 0 without computation is that x is odd with respect to the y-axis whereas the region and the density are symmetric with respect to the y-axis:(x, y) 􀀡 (−x, y) preserves the half annulus and 􀀎 (x, y) = 􀀎 (−x, y) .  \n3. a) Nx = −12y = My , hence F is conservative.  \nb) fx = 3x2 − 6y2 → f = x3 − 6y2x + c(y) → fy = −12xy + c􀀰 (y) = −12xy + 4y. Soc􀀰 (y) = 4y , thus c (y) = 2y2 (+ constant) . In conclusion  \nf = x3 − 6xy2 + 2y2 (+ constant) .  \nc) The curve C starts at (1 , 0) and ends at (1 , 1), therefore  \n􀁚C F · dr = f (1, 1) − f (1, 0) = (1 − 6 + 2) − 1 = −4 .  \n4. a) The parametrization of the circle C is x = cost, y = sint, for 0 􀀔 t \u003C 2􀀙; then dx = − sintdt, dy = costdt and  \nW = 􀁚02􀀙 (5 cost + 3 sint)(− sint)dt + (1 + cos(sint)) costdt.  \nb) Let R be the unit disc inside C;  \n􀁉C F · dr = 􀁚 􀁚R (Nx − My)dA = 􀁚 􀁚R (0 − 3)dA = −3 area(R) = −3􀀙 .  \n5. a) (0 , 4) •􀁯 C3  (1 , 4)  \nC4 C2  \n(0 , 0) ~~ ~~C1~~ 􀀯~~• (1 , 0)  \n􀁉C F · dr = 􀁚 􀁚R (Nx − My)dxdy  \n= 􀁚 􀁚R (y + cos x cosy − cos x cosy)dxdy = 􀁚= 􀁚04 􀁚01 ydxdy = 􀁚04 ydy = [y2 /2]40 = 8.  \n􀁚R ydxdy  \nb) On C4 , F = sin y i, whereas dr is parallel to j. Hence F · dr = 0. Therefore the work of F along C4 equals 0. Thus  \n􀁚C1 +C2 +C3 F · dr = 􀁉C F · dr − 􀁚C4 F · dr = 􀁉C F · dr ; and the total work along C1 + C2 + C3 is equal to the work along C.  \n􀀌  \n6. Let u = 2x − y and v = x + y − 1. The Jacobian 􀀌􀀌  \nHence dudv = 3dxdy and dxdy = 13dudv , so that  \nux uy  \nvx vy  \n􀀌 􀀌  \n􀀌 􀀌  \n􀀌 = 􀀌  \n􀀌 􀀌  \n2 1  \nV = 􀁚 􀁚(2x−y)2 +(x+y)2 4x − y)2 − (x + y − 1)2 dxdy  \n= 􀁚 􀁚u2 +v24− u2 − v2 ) 13dudv  \n= 􀁚02􀀙 􀁚02 (4 − r2 ) 13rdrd􀀒 = 􀁚02􀀙 􀀔 23r2 − 112 r4􀀕 20  \n= 􀁚02􀀙 43 d􀀒 = 8􀀙3 .  \n−1  \n1  \n􀀌  \n􀀌 = 3.  \n􀀌","cbCaie9OQbTSOTSi","https://ap.wps.com/l/cbCaie9OQbTSOTSi","pdf",77535,"English","# Problem 1\n## Draw and transform the region of integration\n# Problem 2\n## Mass of the upper half of an annulus\n## Center of mass integral and symmetry\n# Problem 3\n## Conservative vector field and potential function\n## Line integral over a parametric curve\n# Problem 4\n## Work around the unit circle\n## Green’s theorem evaluation\n# Problem 5\n## Work around a rectangular region boundary\n## Compare partial boundary segments\n# Problem 6\n## Volume enclosed by two surfaces using change of variables","[{\"question\":\"How do you exchange the order of integration for Problem 1?\",\"answer\":\"You redraw the integration region for the original bounds and then restate the double integral in the dxdy order, which splits into two pieces as indicated by the warning.\"},{\"question\":\"Why does the center of mass x-coordinate ̅x equal 0 in Problem 2b?\",\"answer\":\"The integrand xρ is odd with respect to the y-axis, while both the region and density are symmetric under (x,y) → (-x,y), causing the integral to cancel to zero.\"},{\"question\":\"How is Green’s theorem used in Problem 4b?\",\"answer\":\"The line integral around the unit circle is converted to a double integral over the unit disk using Green’s theorem, replacing (N x - M y) with the corresponding expression in the integrand.\"}]","18.02 Practice Exam 3B - Solutions | PDF"]