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Pembuktian keliling lingkaran dari luas lingkaran dengan integral

Seperti yang telah kita ketahui bersama bahwa, keliling lingkaran memiliki rumus

Keliling = 2 π r   ; r = jari-jari
atau
Keliling =  π D    ; D = Diameter

Kita juga tahu bahwa rumus dari luas lingkaran adalah π r2

Nah, sekarang kita mau membuktikan bahwa rumus keliling lingkaran merupakan turunan dari luas lingkaran atau sebaliknya luas lingkaran merupakan integral dari keliling lingkaran.

OK, kita mulai :D

Luas = ∫ Keliling dr ; integral keliling lingkaran terhadap dr

Luas = ∫ 2 π r  dr  ; kita keluarkan 2 π-nya

Luas = 2 π ∫ r  dr  ; Integralkan jari-jarinya terhadap r

Luas = 2 π { 1/2 r}    ; kalikan 1/2 dengan 2

Luas = π r2  

Selesai, mudah bukan :D

Penulis :
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35 Comments

  1. Syati S

    July 10, 2009 at 3:27 am

    Mas sangdedy saya minta posting tentang aljabar matematik, dan matematika diskret dong, bisa nga? mohon bantuannya , terima kasih.

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  2. sangdedi

    sangdedi

    July 10, 2009 at 7:43 am

    aljabarnya bisa tolong diperinci gak ??
    maunya ttg apa ?? jadi saya bisa nulisnya scara lebih detil :-)
    trus discrete math bukannya silabusnya anak kuliahan yah ?? :D

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  3. dewi

    July 23, 2009 at 11:36 am

    :-) :x :cry: :evil: :twisted: :?: :?: :idea: :mrgreen: :arrow: :twisted:

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  4. hangga

    September 9, 2009 at 6:39 am

    sangdedi, bisa gak jabarkan dari awal, dari mana dapatnya rumus volume limas ??
    trims

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  5. sangdedi

    sangdedi

    September 9, 2009 at 3:23 pm

    @ hangga : coba check tulisan saya yang ini Pembuktian Volume Kerucut dengan Integral
    kan kerucut sama limas mirip2 kan yah ?? :D

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  6. irfan

    November 9, 2009 at 3:15 pm

    :D kang gmn cr rumus yg paling mudah untuk menghitung debit

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  7. steven

    November 10, 2009 at 11:57 pm

    makasih ya rumusnya :lol:

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  8. ferna

    December 9, 2009 at 10:19 am

    :-) makasih,penjekasan kamu simple dan mudah dipahami..anak ku pasti cepat paham karna penjelasannya pakai logika sehari-hari

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  9. Steven

    December 17, 2009 at 1:02 pm

    Ini baru kerennzzz… :!:
    Thanks yach… :lol:

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  10. iqiews

    December 24, 2009 at 2:42 am

    kang posting juga tentang Materi Kalkulus dong :!: :D

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  11. sangdedi

    sangdedi

    December 25, 2009 at 6:39 pm

    ok, mau materi ttg apa ??
    differensial ?? integral ?? sebut aja :D

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  12. Michael

    January 2, 2010 at 12:45 am

    :D Makasih ya ^_^ dah banyak mbantu ni. (^^^) :putnam: add fbku ya. sfrzl20_rsbi@yahoo.co.id

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  13. Julaika

    January 2, 2010 at 1:10 am

    Thanks. Sangat membantu untuk sekarang, pel. Matematika. :-)
    Bagi temen” yg mau jd temen FB, add pyutz_cute@yahoo.com atau search aj Zulaycha Putri.
    Makasihh. :D

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  14. Zulaycha

    January 2, 2010 at 1:11 am

    hanks. Sangat membantu untuk sekarang, pel. Matematika. :-)
    Bagi temen” yg mau jd temen FB, add pyutz_cute@yahoo.com atau search aj Zulaycha Putri.
    Makasihh. :D

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  15. Ragushaa

    January 2, 2010 at 1:17 am

    good job aja degh buat mas sangdedi………kern”..

    hagz….hagz

    :lol: :P :D :!:

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  16. vania

    January 9, 2010 at 7:56 am

    om sangdedi . tolong jelasiin pembuktian rumus lingkaraannya dongg ~

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  17. sangdedi

    sangdedi

    January 9, 2010 at 7:12 pm

    itu kan diatas udah ?? 8O
    pembuktian kayak gimana lagi maksudnya ??

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  18. ersa

    January 13, 2010 at 2:18 am

    :arrow: :arrow: :arrow: :arrow: :arrow:

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  19. mhjjhh

    January 16, 2010 at 11:48 am

    :arrow: :lol: 8O :( :-) :-) :D :D :-) 8O :? 8) :x :P :oops: :evil: :cry: :evil: :twisted: :wink: :!: :?: :idea: :arrow: :mrgreen: :idea: :!: :!: :roll: :twisted: :oops: :x :lol: 8)

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  20. andrew

    January 21, 2010 at 11:33 am

    tolong tanya
    Luas = ∫ Keliling dr ; integral keliling lingkaran terhadap dr
    dr itu maksudnya apa?

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  21. sangdedi

    sangdedi

    January 21, 2010 at 4:55 pm

    @andrew : itu artinya diintegralkan terhadap jari2 (r=jari2) :wink:

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  22. Nhyna

    January 31, 2010 at 1:49 am

    Ap nCh ??? :( :o :D

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  23. Ahmad Nurhalim

    May 6, 2010 at 6:13 am

    Wah kebetulan skali..dulu sy alergi matik,jd skrg pas mau belajar jd susah..khusus untk luas n kliling lingkaran,dah lama sy penasaran dgn logika dasarx..syukron infox mas

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  24. sangdedi

    sangdedi

    October 1, 2010 at 7:44 am

    @ahmad : logikanya yang kayak gmn yah mas ?? bisa tolong diperjelas maksudnya ?? :-)

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  25. fhuuer-man

    October 2, 2010 at 5:51 am

    integral itu sebenernya gampang,,yang penting paham apa yang di perintahin di awal kalimat pertanyaan,,gampang dch asal mau latihan truss,, :oops:

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  26. novi

    October 12, 2010 at 2:00 pm

    Makaciii… but membingungkan. :P :roll: :idea: :!: :lol: 8) :arrow:

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  27. sangdedi

    sangdedi

    October 17, 2010 at 5:51 pm

    @firman : terima kasih atas komentarnya yah :wink:
    @novi : tuh kayak yang dibilang firman, sebenernya integral tuh gampang :-)

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  28. maya paembonan

    November 23, 2010 at 11:05 pm

    Bs tolong jelasin asal mula pi g? n asl mula rums luas segitiga…Tq B4 :-)

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  29. eka supratiwi

    March 11, 2011 at 10:57 am

    :wink: :arrow: 8) trus kmbangkn ya………………..

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  30. eka supratiwi

    March 11, 2011 at 10:58 am

    :arrow:

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  31. lukman

    July 20, 2011 at 5:38 pm

    gile nih manteb bener.. cara bahasanya juga enak dibaca bro… jngn bosan2 ya bro untuk berbagi

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  32. aku tdk mengerti

    July 26, 2011 at 7:07 am

    :oops:

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  33. Haris

    September 17, 2011 at 11:18 am

    boleh juga tuh rumusnya :!:

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  34. Rindra

    December 17, 2011 at 2:45 pm

    contoh penerapan integral dalam bidang teknik sipil apa yaa??? :-)

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  35. Ony

    December 26, 2013 at 12:58 am

    Mantap Om Dedi…
    Kapan-kapan mau sumbang tulisan juga ah…

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