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Volume benda putar yang terbentuk dari daerah di kuadran I yang dibatasi oleh kurva begin mathsize 14px style x equals 2 square root of 3 y squared end style, sumbu Y, dan lingkaran begin mathsize 14px style x squared plus y squared equals 1 end style, diputar mengelilingi sumbu Y adalah ....

 

  1. begin mathsize 14px style 4 over 60 straight pi space satuan space volume end style

  2. begin mathsize 14px style 17 over 60 straight pi space satuan space volume end style

  3. begin mathsize 14px style 23 over 60 straight pi space satuan space volume end style

  4. begin mathsize 14px style 44 over 60 straight pi space satuan space volume end style

  5. begin mathsize 14px style 112 over 60 straight pi space satuan space volume end style

S. Intan

Master Teacher

Mahasiswa/Alumni Institut Pertanian Bogor

Jawaban terverifikasi

Pembahasan

begin mathsize 14px style Volume space benda space putar space pada space sumbu space straight y.  bold Kurva bold space bold I  straight x equals 2 square root of 3 straight space straight y squared  straight x squared equals 12 straight space straight y to the power of 4    bold Kurva bold space bold II  straight x squared plus straight y squared equals 1  straight x squared equals 1 minus straight y squared    Tentukan space titik space potongnya space terlebih space dulu  12 straight y to the power of 4 equals 1 minus straight y squared  12 straight y to the power of 4 plus straight y squared minus 1 equals 0  open parentheses 3 straight y squared plus 1 close parentheses open parentheses 4 straight y squared minus 1 close parentheses equals 0  straight y squared equals negative 1 third space atau space straight y squared equals 1 fourth  Untuk space straight y squared equals negative 1 third comma space tidak space ada space nilai space straight y space yang space memenuhi.  Ambil space straight y squared equals 1 fourth rightwards arrow straight y equals plus-or-minus 1 half  end style

begin mathsize 14px style Terlihat space batas minus batasnya space adalah space 0 space sampai space ½ space space untuk space straight x equals 2 square root of 3 straight space straight y squared space dan  ½ space space sampai space 1 space untuk space straight x squared plus straight y squared equals 1 comma space sehingga space volumenya colon  straight V equals straight pi open square brackets integral subscript 0 superscript begin inline style 1 half end style end superscript space straight x subscript 1 superscript 2 space dy plus integral subscript begin inline style 1 half end style end subscript superscript 1 space straight x subscript 2 superscript 2 space dy close square brackets  straight V equals straight pi open square brackets integral subscript 0 superscript begin inline style 1 half end style end superscript space 12 straight y to the power of 4 space dy plus integral subscript begin inline style 1 half end style end subscript superscript 1 space left parenthesis 1 minus straight y squared right parenthesis space dy close square brackets  straight V equals straight pi open vertical bar 12 over 5 straight y to the power of 5 close vertical bar subscript 0 superscript begin inline style 1 half end style end superscript plus straight pi open vertical bar straight y minus 1 third straight y cubed close vertical bar subscript begin inline style 1 half end style end subscript superscript 1  straight V equals straight pi open curly brackets open parentheses 12 over 5 open parentheses 1 half close parentheses to the power of 5 close parentheses plus open parentheses open parentheses 1 close parentheses minus 1 third open parentheses 1 close parentheses cubed close parentheses minus open parentheses open parentheses 1 half close parentheses minus 1 third open parentheses 1 half close parentheses cubed close parentheses close curly brackets  straight V equals straight pi open curly brackets 3 over 40 plus open parentheses 2 over 3 minus 11 over 24 close parentheses close curly brackets  straight V equals straight pi open parentheses 3 over 40 plus 5 over 24 close parentheses  straight V equals 17 over 60 straight pi space satuan space volume end style

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