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Pertanyaan

Daerah R dibatasi oleh y equals b square root of x comma y equals b x comma untuk x element of open square brackets 0 comma 2 close square brackets. Jika volume benda padat yang didapat dengan memutar R terhadap sumbu x adalah straight pi, maka b = ....

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A. Rizky

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

Pembahasan

straight y equals straight y space  straight b square root of straight x equals bx space  square root of straight x equals straight x space  straight x minus square root of straight x equals 0 space  square root of straight x space open parentheses square root of straight x minus 1 close parentheses equals 0    square root of straight x equals 0 rightwards arrow box enclose straight x equals 0 end enclose  Dan  square root of straight x minus 1 equals 0 rightwards arrow square root of straight x equals 1 rightwards arrow box enclose straight x equals 1 end enclose    Maka space kurva space dan space garis space akan space berpotongan space di space titik space straight x equals 0 space dan space straight x equals 1  seperti space ditunjukkan space oleh space gambar space berikut space ini space colon

Catatan space colon thin space daerah space yang space diarsir space menunjukkan space daerah space straight R.    Volume space benda space padat space pada space interval space 0 less or equal than straight x less or equal than 1 space adalah  straight V subscript 1 equals straight pi integral from 0 to 1 of open parentheses straight b square root of straight x close parentheses squared minus open parentheses bx close parentheses squared space dx equals straight pi integral from 0 to 1 of straight b squared straight x minus straight b squared straight x squared space dx equals straight pi open open parentheses straight b squared over 2 straight x squared minus straight b squared over 3 straight x cubed close parentheses close square brackets subscript 0 superscript 1  equals straight pi open square brackets open parentheses straight b squared over 2 open parentheses 1 close parentheses squared minus straight b squared over 3 open parentheses 1 close parentheses cubed close parentheses minus open parentheses straight b squared over 2 open parentheses 0 close parentheses squared minus straight b squared over 3 open parentheses 0 close parentheses cubed close parentheses close square brackets equals straight pi open square brackets open parentheses straight b squared over 2 minus straight b squared over 3 close parentheses minus open parentheses 0 minus 0 close parentheses close square brackets equals box enclose fraction numerator straight b squared straight pi over denominator 6 end fraction end enclose  Sedangkan space volume space benda space padat space pada space interval space 1 less than straight x less or equal than 2 space adalah  straight V subscript 2 equals straight pi integral from 1 to 2 of open parentheses bx close parentheses squared minus open parentheses straight b square root of straight x close parentheses squared dx equals straight pi integral from 1 to 2 of straight b squared straight x squared minus straight b squared straight x space dx equals straight pi open open parentheses straight b squared over 3 straight x cubed minus straight b squared over 2 straight x squared close parentheses close square brackets subscript 1 superscript 2  equals straight pi open square brackets open parentheses straight b squared over 3 open parentheses 2 close parentheses cubed minus straight b squared over 2 open parentheses 2 close parentheses squared close parentheses minus open parentheses straight b squared over 3 open parentheses 1 close parentheses cubed minus straight b squared over 2 open parentheses 1 close parentheses squared close parentheses close square brackets equals straight pi open square brackets open parentheses fraction numerator 8 straight b squared over denominator 3 end fraction minus 2 straight b squared close parentheses minus open parentheses straight b squared over 3 minus straight b squared over 2 close parentheses close square brackets  equals straight pi open square brackets fraction numerator 2 straight b squared over denominator 3 end fraction minus open parentheses negative straight b squared over 6 close parentheses close square brackets equals straight pi open parentheses fraction numerator 2 straight b squared over denominator 3 end fraction plus straight b squared over 6 close parentheses equals box enclose fraction numerator 5 straight b squared straight pi over denominator 6 end fraction end enclose  Sehingga space volume space benda space padat space pada space interval space 0 less or equal than straight x less or equal than 2 space adalah  straight V subscript 1 plus straight V subscript 2 equals fraction numerator straight b squared straight pi over denominator 6 end fraction plus fraction numerator 5 straight b squared straight pi over denominator 6 end fraction equals box enclose straight b squared straight pi end enclose    Dari space perhitungan space didapatkan space volume space yaitu space box enclose straight b squared straight pi end enclose.  Dari space soal space didapatkan space volume space yaitu space box enclose straight pi.  Maka  straight b squared straight pi equals straight pi  straight b squared equals 1  box enclose straight b equals plus-or-minus 1 end enclose

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