Calculus: The Logical Extension of Arithmetic

Infinity Times Zero and the Concept of Integration

Author(s): Seymour B. Elk

Pp: 186-261 (76)

DOI: 10.2174/9781681082035116010008

* (Excluding Mailing and Handling)

Abstract

In this chapter, the second of the processes in the traditional sequence of courses that comprise the calculus curriculum, “integral calculus”, is examined in terms of its algebraic foundation. A new paradigm is proposed that views “integration”, also often designated as “anti- (or inverse) differentiation”, as a form of infinity multiplied by zero. Such a protocol is demonstrated as being the inverse operation to the previously developed process of differentiation in Chapter 3. However, unlike the function produced by the process of differentiation, this inverse is either not unique and needs to be supplemented with an arbitrary unspecified constant (which is addended to the generated function) or a set of limits. Along with “sloughing through” several of the techniques associated with such anti-differentiation, the mathematical underpinning of this inverse operation is introduced. This is then supplemented by an expansion of the horizon of “what is mathematics?” to introduce (1) a special (more advanced) function, called the Dirac delta function, which, in an altogether different manner is also subsumed by the over-arching concept of infinity multiplied by zero, and (2) the theoretical base from one limited to continuous functions (called “Riemann integration”) to a larger set that includes selected discontinuous functions (called “Lebesgue integration”).


Keywords: Anti-Derivatives, Area of Surface of Revolution, Dirac Delta, Direct Substitutions, Indirect Substitutions, Integral Calculus, Integration By Parts, Kronecker Delta, Lebesque Integral, Length of Planar Curves, Volume of Revolution (Slices vs. Shells).

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      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • (select j.user_access_id,j.volume_id,j.user_id,j.from_ip,j.to_ip,j.type,j.track,"Trial Access" as access_base_text, CASE j.type WHEN 111 THEN true ELSE false END as terms_popup, "S" as content_download_type, null as restricted_user_access_key, null as restricted_user_access_type, CASE j.track WHEN 0 THEN false ELSE true END as tracksAccessByVolume,"chapter" as access_level, j.chapter_id from `user_access_chapter_trail` as `j` inner join `user_access_info` as `info` on `j`.`user_access_id` = `info`.`user_access_id` where ((j.from_ip <= 314564252 and j.to_ip >=314564252 )) and ((j.from_date <= 1740473382 and j.to_date >= 1740473382 and j.perpetual= 0) or (j.from_date is null and j.to_date is null and j.perpetual= 1)) and j.chapter_id = 9751) union (select j.user_access_id,j.volume_id,j.user_id,j.from_ip,j.to_ip,j.type,j.track,"Subscribed" as access_base_text, false as terms_popup, "S" as content_download_type, null as restricted_user_access_key, null as restricted_user_access_type, CASE j.track WHEN 0 THEN false ELSE true END as tracksAccessByVolume,"chapter" as access_level, j.chapter_id from `user_access_chapter_corporate` as `j` inner join `user_access_info` as `info` on `j`.`user_access_id` = `info`.`user_access_id` where ((j.from_ip <= 314564252 and j.to_ip >=314564252 )) and ((j.from_date <= 1740473382 and j.to_date >= 1740473382 and j.perpetual= 0) or (j.from_date is null and j.to_date is null and j.perpetual= 1)) and j.chapter_id = 9751) union (select j.user_access_id,j.volume_id,j.user_id,j.from_ip,j.to_ip,j.type,j.track,"Token Based Access" as access_base_text, false as terms_popup, "R" as content_download_type, j.uc_id as restricted_user_access_key, "v" as restricted_user_access_type, CASE j.track WHEN 0 THEN false ELSE true END as tracksAccessByVolume,"chapter" as access_level, j.chapter_id from `user_access_chapter_token` as `j` inner join `user_access_info` as `info` on `j`.`user_access_id` = `info`.`user_access_id` where ((j.from_ip <= 314564252 and j.to_ip >=314564252 )) and ((j.from_date <= 1740473382 and j.to_date >= 1740473382 and j.perpetual= 0) or (j.from_date is null and j.to_date is null and j.perpetual= 1)) and j.chapter_id = 9751)
      1.03msalphaeurekaselec_live_10_06_2022UserAccess.php#1266
      Backtrace
      • 13. app/Models/UserAccess/UserAccess.php:1266
      • 14. app/Http/Controllers/ChapterController.php:1549
      • 15. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
      • 16. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
      • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
    • select b.banner_link, b.banner_name, b.banner_script, b.banner_remarks, b.banner_file_name, b.banner_placement, bd.content_type, bd.content_id from `tbl_banner` as `b` inner join `tbl_banner_detail` as `bd` on `bd`.`banner_id` = `b`.`banner_id` inner join `users` as `u` on `u`.`id` = `b`.`updated_by` where `b`.`banner_ebook` = 1 and `bd`.`content_type` = 'B' and `bd`.`content_id` = 2186 and `b`.`banner_from_date` <= 1740473382 and `b`.`banner_to_date` >= 1740473382 and `b`.`banner_status` = 'A' order by `b`.`banner_id` desc
      1.58msalphaeurekaselec_live_10_06_2022Banner.php#95
      Bindings
      • 0: 1
      • 1: B
      • 2: 2186
      • 3: 1740473382
      • 4: 1740473382
      • 5: A
      Backtrace
      • 13. app/Models/Banner.php:95
      • 14. app/Http/helpers.php:404
      • 17. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 18. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    • select count(*) as aggregate from `uc_cart_products` where cart_id='7f06a7fb433fbfb947578cb16061da9c'
      410μsalphaeurekaselec_live_10_06_2022Cart.php#271
      Backtrace
      • 15. app/Models/Cart.php:271
      • 16. view::layouts._header:260
      • 18. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:124
      • 19. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
      • 20. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:73
    App\Models\Keywords
    11Keywords.php#?
    App\Models\Chapter
    1Chapter.php#?
    App\Models\Ebook
    1Ebook.php#?
        _token
        vOAEdkHRhh24QwQ99omvxPLiwjZzGWtFJD8mNDNG
        uc_cart_id
        7f06a7fb433fbfb947578cb16061da9c
        _previous
        array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/9751" ]
        _flash
        array:2 [ "old" => [] "new" => [] ]
        PHPDEBUGBAR_STACK_DATA
        []
        path_info
        /chapter/9751
        status_code
        200
        
        status_text
        OK
        format
        html
        content_type
        text/html; charset=UTF-8
        request_query
        []
        
        request_request
        []
        
        request_headers
        0 of 0
        array:17 [ "x-https" => array:1 [ 0 => "1" ] "host" => array:1 [ 0 => "alpha.eurekaselect.com" ] "priority" => array:1 [ 0 => "u=0, i" ] "cookie" => array:1 [ 0 => "wayfindertarget=eyJpdiI6Ino5NVRxcFY4TUM3Q3g1aGwxRkJ5TXc9PSIsInZhbHVlIjoiTEJaUm9CbWc2ZEVmRG9pRUZ3WHh0UFcySXViNDh4TWZoY1B6OVpaLzBUbz0iLCJtYWMiOiJiZDI2MmM4ZDUwMDdlMTJjMGY3YWFmZGEzZjkxYTZlYTY2YTIzNzhkNjg4ZDQ5NTI2MGQ5ZTg1NjI1OTlkZTI2IiwidGFnIjoiIn0%3Dwayfindertarget=eyJpdiI6Ino5NVRxcFY4TUM3Q3g1aGwxRkJ5TXc9PSIsInZhbHVlIjoiTEJaUm9CbWc2ZEVmRG9pRUZ3WHh0UFcySXViNDh4TWZoY1B6OVpaLzBUbz0iLCJtYWMiOiJiZDI2MmM4ZDUwMDdl" ] "accept-encoding" => array:1 [ 0 => "gzip, deflate, br, zstd" ] "sec-fetch-dest" => array:1 [ 0 => "document" ] "sec-fetch-user" => array:1 [ 0 => "?1" ] "sec-fetch-mode" => array:1 [ 0 => "navigate" ] "sec-fetch-site" => array:1 [ 0 => "none" ] "accept" => array:1 [ 0 => "text/html,application/xhtml+xml,application/xml;q=0.9,image/avif,image/webp,image/apng,*/*;q=0.8,application/signed-exchange;v=b3;q=0.7" ] "user-agent" => array:1 [ 0 => "Mozilla/5.0 AppleWebKit/537.36 (KHTML, like Gecko; compatible; ClaudeBot/1.0; +claudebot@anthropic.com)" ] "upgrade-insecure-requests" => array:1 [ 0 => "1" ] "sec-ch-ua-platform" => array:1 [ 0 => ""Windows"" ] "sec-ch-ua-mobile" => array:1 [ 0 => "?0" ] "sec-ch-ua" => array:1 [ 0 => ""HeadlessChrome";v="129", "Not=A?Brand";v="8", "Chromium";v="129"" ] "cache-control" => array:1 [ 0 => "no-cache" ] "pragma" => array:1 [ 0 => "no-cache" ] ]
        request_cookies
        0 of 0
        array:1 [ "wayfindertarget" => null ]
        response_headers
        0 of 0
        array:5 [ "content-type" => array:1 [ 0 => "text/html; charset=UTF-8" ] "cache-control" => array:1 [ 0 => "no-cache, private" ] "date" => array:1 [ 0 => "Tue, 25 Feb 2025 08:49:42 GMT" ] "set-cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6IjlzQnpVM3FNdHlCYnBtRTJNUUdMZHc9PSIsInZhbHVlIjoiclk5WG11bk1oOFZ5cnVpZDAvZUlmYXBTQTd1VnJtcVFwVE03cWdHQ3h4Rkthb1ZsbExBdEhyNXBlcFNWVnpRRzA4TFNDVC84Ym9TZWpSVDVwYTBPYkxQQWZWbmc1ams5MXRzYjBZMEVnRkRBSXI0WjMyZHpKYVBQRUE2WnYwTEEiLCJtYWMiOiI3NjhlZjM2ZDhkYjVkYWUzNDFhYTljYmI3YTY3MTdiNDFiNDE1MThiNTBlZGE0ZTdjNDA4MTU1YjEwMGQ2MDY4IiwidGFnIjoiIn0%3D; expires=Thu, 27 Feb 2025 20:49:42 GMT; Max-Age=216000; path=/; secure; samesite=laxXSRF-TOKEN=eyJpdiI6IjlzQnpVM3FNdHlCYnBtRTJNUUdMZHc9PSIsInZhbHVlIjoiclk5WG11bk1oOFZ5cnVpZDAvZUlmYXBTQTd1VnJtcVFwVE03cWdHQ3h4Rkthb1ZsbExBdEhyNXBlcFNWVnpRRzA4TFNDV" 1 => "alphaeurekaselectnet_session=eyJpdiI6IlJHZ2dwYW9PYkJETnAxb3V3Zlo1YWc9PSIsInZhbHVlIjoibW03WFphL2Nic2VTcFNTejBqVktDUVFNb2xmNmhIMVVuMjlKUVhyWVVodDRsNExPbkNrTW14NXo1bmRIekdpWDMvQ29OQ1JRcnJiNXVldGxzdFIrYmhSZC9jc1VSNW5TUjJxZVRMTWtlLzFYUE8yeFFES1BuMUszVnVwV0tjSHYiLCJtYWMiOiI2YmQ3MWYwNGM2YTg2ZTYzZTFhNDdmMTIzYjAzYzM5YjZlYmZhMDZjMjI3NWQwMmQyZWNmYzcwOTg5ODVhNmZlIiwidGFnIjoiIn0%3D; path=/; secure; httponly; samesite=laxalphaeurekaselectnet_session=eyJpdiI6IlJHZ2dwYW9PYkJETnAxb3V3Zlo1YWc9PSIsInZhbHVlIjoibW03WFphL2Nic2VTcFNTejBqVktDUVFNb2xmNmhIMVVuMjlKUVhyWVVodDRsNExPbkNrTW14NXo" ] "Set-Cookie" => array:2 [ 0 => "XSRF-TOKEN=eyJpdiI6IjlzQnpVM3FNdHlCYnBtRTJNUUdMZHc9PSIsInZhbHVlIjoiclk5WG11bk1oOFZ5cnVpZDAvZUlmYXBTQTd1VnJtcVFwVE03cWdHQ3h4Rkthb1ZsbExBdEhyNXBlcFNWVnpRRzA4TFNDVC84Ym9TZWpSVDVwYTBPYkxQQWZWbmc1ams5MXRzYjBZMEVnRkRBSXI0WjMyZHpKYVBQRUE2WnYwTEEiLCJtYWMiOiI3NjhlZjM2ZDhkYjVkYWUzNDFhYTljYmI3YTY3MTdiNDFiNDE1MThiNTBlZGE0ZTdjNDA4MTU1YjEwMGQ2MDY4IiwidGFnIjoiIn0%3D; expires=Thu, 27-Feb-2025 20:49:42 GMT; path=/; secureXSRF-TOKEN=eyJpdiI6IjlzQnpVM3FNdHlCYnBtRTJNUUdMZHc9PSIsInZhbHVlIjoiclk5WG11bk1oOFZ5cnVpZDAvZUlmYXBTQTd1VnJtcVFwVE03cWdHQ3h4Rkthb1ZsbExBdEhyNXBlcFNWVnpRRzA4TFNDV" 1 => "alphaeurekaselectnet_session=eyJpdiI6IlJHZ2dwYW9PYkJETnAxb3V3Zlo1YWc9PSIsInZhbHVlIjoibW03WFphL2Nic2VTcFNTejBqVktDUVFNb2xmNmhIMVVuMjlKUVhyWVVodDRsNExPbkNrTW14NXo1bmRIekdpWDMvQ29OQ1JRcnJiNXVldGxzdFIrYmhSZC9jc1VSNW5TUjJxZVRMTWtlLzFYUE8yeFFES1BuMUszVnVwV0tjSHYiLCJtYWMiOiI2YmQ3MWYwNGM2YTg2ZTYzZTFhNDdmMTIzYjAzYzM5YjZlYmZhMDZjMjI3NWQwMmQyZWNmYzcwOTg5ODVhNmZlIiwidGFnIjoiIn0%3D; path=/; secure; httponlyalphaeurekaselectnet_session=eyJpdiI6IlJHZ2dwYW9PYkJETnAxb3V3Zlo1YWc9PSIsInZhbHVlIjoibW03WFphL2Nic2VTcFNTejBqVktDUVFNb2xmNmhIMVVuMjlKUVhyWVVodDRsNExPbkNrTW14NXo" ] ]
        session_attributes
        0 of 0
        array:5 [ "_token" => "vOAEdkHRhh24QwQ99omvxPLiwjZzGWtFJD8mNDNG" "uc_cart_id" => "7f06a7fb433fbfb947578cb16061da9c" "_previous" => array:1 [ "url" => "https://alpha.eurekaselect.com/chapter/9751" ] "_flash" => array:2 [ "old" => [] "new" => [] ] "PHPDEBUGBAR_STACK_DATA" => [] ]
        ClearShow all
        Date ↕MethodURLData
        #12025-02-25 08:49:42GET/chapter/97511121813