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Daerah R dibatasi oleh , dan sumbu x. Volume benda...

Daerah R dibatasi oleh y equals square root of x comma space y equals negative x plus 6, dan sumbu x. Volume benda padat yang didapat dengan memutar R terhadap sumbu x adalah...

  1. fraction numerator 8 straight pi over denominator 3 end fraction

  2. fraction numerator 16 straight pi over denominator 3 end fraction

  3. fraction numerator 24 straight pi over denominator 3 end fraction

  4. fraction numerator 32 straight pi over denominator 3 end fraction

  5. fraction numerator 40 straight pi over denominator 3 end fraction

Jawaban:

straight y equals straight y  square root of straight x equals negative straight x plus 6  straight x equals left parenthesis negative straight x plus 6 right parenthesis squared  straight x equals straight x squared minus 12 straight x plus 36  straight x squared minus 13 straight x plus 36 equals 0  left parenthesis straight x minus 4 right parenthesis left parenthesis straight x minus 9 right parenthesis equals 0  box enclose straight x equals 4 end enclose space atau space box enclose straight x equals 9 end enclose space    Namun space untuk space straight x equals 9 space mengakibatkan space persamaan space square root of straight x equals negative straight x plus 6 space tidak  terpenuhi comma space sehingga space box enclose straight x equals 4 end enclose.    Maka space kurva space dan space garis space akan space berpotongan space di space titik space straight x equals 4 space seperti space ditunjukkan  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 4 space adalah  straight V subscript 1 equals straight pi integral from 0 to 4 of open parentheses square root of straight x close parentheses squared dx equals straight pi integral from 0 to 4 of xdx equals straight pi open open parentheses straight x squared over 2 close parentheses close square brackets subscript 0 superscript 4 equals straight pi open square brackets open parentheses open parentheses 4 close parentheses squared over 2 close parentheses minus open parentheses open parentheses 0 close parentheses squared over 2 close parentheses close square brackets equals straight pi open square brackets 8 minus 0 close square brackets equals box enclose 8 straight pi end enclose  Sedangkan space volume space benda space padat space pada space interval space 4 less than straight x less or equal than 6 space adalah  straight V subscript 2 equals straight pi integral from 4 to 6 of open parentheses negative straight x plus 6 close parentheses squared dx equals straight pi integral from 4 to 6 of open parentheses straight x squared minus 12 straight x plus 36 close parentheses dx equals straight pi open open parentheses 1 third straight x cubed minus 6 straight x squared plus 36 straight x close parentheses close square brackets subscript 4 superscript 6  space space space space equals straight pi open square brackets open parentheses 1 third open parentheses 6 close parentheses cubed minus 6 open parentheses 6 close parentheses squared plus 36 open parentheses 6 close parentheses close parentheses minus open parentheses 1 third open parentheses 4 close parentheses cubed minus 6 open parentheses 4 close parentheses squared plus 36 open parentheses 4 close parentheses close parentheses close square brackets  space space space space equals straight pi open square brackets open parentheses 72 minus 216 plus 216 close parentheses minus open parentheses 64 over 3 minus 96 plus 144 close parentheses close square brackets equals straight pi open parentheses 72 minus 208 over 3 close parentheses equals box enclose 8 over 3 straight pi 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 6 space adalah  straight V subscript 1 plus straight V subscript 2 equals 8 straight pi plus 8 over 3 straight pi equals box enclose 32 over 3 straight pi end enclose

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