[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-297382-105":59,"doc-detail-297382-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","surface-area-of-a-sphere","Surface Area of a Sphere","","The document derives the surface area of a sphere by treating it as a surface of revolution generated from revolving a semicircle of radius a about the x-axis. It sets integration bounds between x1 and x2 with −a ≤ x1 ≤ x2 ≤ a and uses the relationship between y, dy/dx, and the arc-length element ds to form the surface area integral. The final result is 2πa(x2 − x1), with checks for a hemisphere and the full sphere yielding 2πa² and 4πa². It also discusses rotating about the y-axis and clarifies that shell methods compute volumes rather than surface areas.",{"@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/surface-area-of-a-sphere/297382/",{"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/surface-area-of-a-sphere/297382.png","ImageObject",300,407,{"name":92,"@type":93},"Emma Wilson","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-25","2026-09-18",true,{"@type":102,"interactionType":103,"userInteractionCount":44},"InteractionCounter",{"@type":104},"ViewAction",{"@type":106,"mainEntity":107},"FAQPage",[108,114,118],{"name":109,"@type":110,"acceptedAnswer":111},"How is the sphere modeled for the surface-area calculation?","Question",{"text":112,"@type":113},"The sphere is modeled as a surface of revolution formed by revolving a half circle of radius a about the x-axis.","Answer",{"name":115,"@type":110,"acceptedAnswer":116},"What is the resulting formula for the surface area between x1 and x2?",{"text":117,"@type":113},"The derived formula is Area = 2πa(x2 − x1).",{"name":119,"@type":110,"acceptedAnswer":120},"Why does the shell method not directly work for computing surface area here?",{"text":121,"@type":113},"Shells have thickness dx and are used in integrals that compute volumes; they do not directly sum surface areas.","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},297382,1790065696,{"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":44,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":19,"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":143,"read_time":52},3848291630094,"https://eur-avatar.wpscdn.com/davatar_085a072bc5b1113ac321206ff7593b45","Surface Area of a Sphere  \nIn this example we will complete the calculation of the area of a surface of rotation. If we’re going to go to the effort to complete the integral, the answer should be a nice one; one we can remember. It turns out that calculating the surface area of a sphere gives us just such an answer.  \nWe’ll think of our sphere as a surface of revolution formed by revolving a half circle of radius a about the x-axis. We’ll be integrating with respect to x, and we’ll let the bounds on our integral be x1 and x2 with −a ≤ x 1 ≤ x2 ≤ aas sketched in Figure 1.  \nFigure 1: Part of the surface of a sphere.  \nRemember that in an earlier example we computed the length of an infinitesimal segment of a circular arc of radius 1:  \nds = 􀁲 ~~ ~~1~~ ~~x2  dx  \nIn this example we let the radius equal a so that we can see how the surface area depends on the radius. Hence:  \ny = 􀁰a2 − x2  −x   \ny􀀰 =  \n√a2 − x2  \nds = 􀁲1 + a2~~ ~~2~~ ~~x2  dx = 􀁲 a2~~ ~~−a2+x2x2 dx  \n= 􀁲 a2~~ ~~x2 dx.  \nThe formula for the surface area indicated in Figure 1 is: Area = 􀁚xx12 2πy ds  \nds  \ny 􀁺~~  ~~􀁽􀁼~~  ~~􀁻= 􀁚xx12 2π a~~ ~~x 􀁲 a2~~ ~~~~ ~~x2 dx = 􀁚xx12 2π 􀁰 a2 − x2 ~~ ~~√a2a−~~ ~~x2 dx = 􀁚xx12 2πadx  \n= 2πa(x2 − x1 ) .  \nSpecial Cases  \nWhen possible, we should test our results by plugging in values to see if our answer is reasonable. Here, if we set x 1 = 0 and x2 = a we should get the surface area of a hemisphere of radius a:  \nFigure 2: Right hemisphere.  \n2πa(x2 − x1 ) = 2πa(a − 0)= 2πa2  \nWe get the surface area of the whole sphere by letting x 1 = −a and x2 = a:  \n2πa(x2 − x1 ) = 2πa(a − (−a))  \n= 4πa2  \nQuestion: Would it be possible to rotate around the y-axis?  \nAnswer: Yes. If we rotate around the y axis and integrate with respect to x (calculating the surface area of a vertical slice, as we did here) we’d be  \nadding up little strips of area. If we integrate with respect to y and find the surface area between two vertical positions y 1 and y2 we’d get exactly the same calculation.  \nQuestion: Can you compute surface area using shells?  \nAnswer: The short answer is “not quite”. We use the word shell to describe something which has a thickness dx. Shells have volume, integrals which involve shells compute volumes, not surface areas.  \nTo compute surface area you need to sum up the areas of small regions of your surface, but those small regions can have any shape whose area you can measure.  \nMIT OpenCourseWare  \n[http://ocw.mit.edu](http://ocw.mit.edu)  \n18.01SC Single Variable Calculus Fall 2010  \nFor information about citing these materials or our Terms of Use, visit: [http://ocw.mit.edu/terms](http://ocw.mit.edu/terms).","cbCaiif0R3jJtXXZ","https://ap.wps.com/l/cbCaiif0R3jJtXXZ","pdf",169843,"English","# Surface Area Derivation\n## Surface of Revolution Setup\n## Surface Area Integral and Simplification\n## Special Cases\n## Alternative Rotation and Shell Method Notes","[{\"question\":\"How is the sphere modeled for the surface-area calculation?\",\"answer\":\"The sphere is modeled as a surface of revolution formed by revolving a half circle of radius a about the x-axis.\"},{\"question\":\"What is the resulting formula for the surface area between x1 and x2?\",\"answer\":\"The derived formula is Area = 2πa(x2 − x1).\"},{\"question\":\"Why does the shell method not directly work for computing surface area here?\",\"answer\":\"Shells have thickness dx and are used in integrals that compute volumes; they do not directly sum surface areas.\"}]","Surface Area of a Sphere | PDF",1789744174]