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Pertanyaan

Daerah R dibatasi oleh y equals a square root of x comma space y equals a x squared untuk x element of left square bracket 0 comma 2 right square bracket. Jika volume benda padat yang didapat dengan memutar R terhadap sumbu x adalah 5 straight pi, maka a = ....

  1. -5

  2. -4

  3. -3

  4. -2

  5. -1

A. Rizky

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

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Pembahasan

 

straight y equals straight y  straight a square root of straight x equals ax squared  square root of straight x equals straight x squared  straight x squared minus square root of straight x equals 0  square root of straight x open parentheses straight x square root of straight x minus 1 close parentheses equals 0  Maka  square root of straight x equals 0 rightwards arrow box enclose straight x equals 0 end enclose  Dan  straight x square root of straight x minus 1 equals 0 rightwards arrow straight x 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 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 a square root of straight x close parentheses squared minus open parentheses ax squared close parentheses squared dx equals straight pi integral from 0 to 1 of straight a squared straight x minus straight a squared straight x to the power of 4 dx equals straight pi open open parentheses straight a squared over 2 straight x squared minus straight a squared over 5 straight x to the power of 5 close parentheses close square brackets subscript 0 superscript 1  equals straight pi open square brackets open parentheses straight a squared over 2 open parentheses 1 close parentheses squared minus straight a squared over 5 open parentheses 1 close parentheses to the power of 5 close parentheses minus open parentheses straight a squared over 2 open parentheses 0 close parentheses squared minus straight a squared over 5 open parentheses 0 close parentheses to the power of 5 close parentheses close square brackets equals straight pi open square brackets open parentheses straight a squared over 2 minus straight a squared over 5 close parentheses minus open parentheses 0 minus 0 close parentheses close square brackets equals box enclose fraction numerator 3 straight a squared straight pi over denominator 10 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 ax squared close parentheses squared minus open parentheses straight a square root of straight x close parentheses squared dx equals straight pi integral from 1 to 2 of straight a squared straight x to the power of 4 minus straight a squared straight x space dx equals straight pi open open parentheses straight a squared over 5 straight x to the power of 5 minus straight a squared over 2 straight x squared close parentheses close square brackets subscript 1 superscript 2  equals straight pi open square brackets open parentheses straight a squared over 5 open parentheses 2 close parentheses to the power of 5 minus straight a squared over 2 open parentheses 2 close parentheses squared close parentheses minus open parentheses straight a squared over 5 open parentheses 1 close parentheses to the power of 5 minus straight a squared over 2 open parentheses 1 close parentheses squared close parentheses close square brackets equals straight pi open square brackets open parentheses fraction numerator 32 straight a squared over denominator 5 end fraction minus 2 straight a squared close parentheses minus open parentheses straight a squared over 5 minus straight a squared over 2 close parentheses close square brackets  equals straight pi open square brackets fraction numerator 22 straight a squared over denominator 5 end fraction minus open parentheses negative fraction numerator 3 straight a squared over denominator 10 end fraction close parentheses close square brackets equals straight pi open parentheses fraction numerator 22 straight a squared over denominator 5 end fraction plus fraction numerator 3 straight a squared over denominator 10 end fraction close parentheses equals box enclose fraction numerator 47 straight a squared straight pi over denominator 10 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 3 straight a squared straight pi over denominator 10 end fraction plus fraction numerator 47 straight a squared straight pi over denominator 10 end fraction equals box enclose 5 straight a squared straight pi end enclose    Dari space soal space didapatkan space volume space yaitu space 5 straight pi.  Maka  5 straight a squared straight pi equals 5 straight pi  straight a squared equals 1  box enclose straight a equals plus-or-minus 1 end enclose

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Daerah R dibatasi oleh y=bx​, y = bx, untuk x ∈ [0,2]. Jika volume benda padat yang didapat dengan memutar R terhadap sumbu x adalah π, maka b = ....

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