Volume benda putar yang terbentuk karena daerah dibatasi y=9−x2 dan y=x+7 diputar 360∘ mengelilingi sumbu x adalah ...

Pertanyaan

Volume benda putar yang terbentuk karena daerah dibatasi y equals 9 minus x squared dan y equals x plus 7 diputar 360 degree mengelilingi sumbu x adalah ...

  1. 40 2 over 3 straight pi

  2. 45 3 over 5 straight pi

  3. 48 4 over 5 straight pi

  4. 53 2 over 5 straight pi

  5. 66 3 over 5 straight pi

B. Hary

Master Teacher

Jawaban terverifikasi

Jawaban

jawaban yang tepat adalah E.

Pembahasan

Mencari titik potong kurva.

table attributes columnalign right center left columnspacing 0px end attributes row y equals y row cell 9 minus x squared end cell equals cell x plus 7 end cell row cell 9 minus x squared minus x minus 7 end cell equals 0 row cell negative x squared minus x plus 2 end cell equals 0 row cell x squared plus x minus 2 end cell equals 0 row cell open parentheses x plus 2 close parentheses open parentheses x minus 1 close parentheses end cell equals 0 end table

Diperoleh:

table attributes columnalign right center left columnspacing 0px end attributes row cell x plus 2 end cell equals 0 row x equals cell negative 2 end cell row blank blank blank row cell x minus 1 end cell equals 0 row x equals 1 end table

Titik potong kurva x equals negative 2 dan x equals 1.

Kemudian tentukan volume benda putarnya.

table attributes columnalign right center left columnspacing 0px end attributes row V equals cell straight pi integral subscript negative 2 end subscript superscript 1 open parentheses 9 minus straight x squared close parentheses squared minus open parentheses straight x plus 7 close parentheses squared d x end cell row blank equals cell straight pi integral subscript negative 2 end subscript superscript 1 81 minus 18 straight x squared plus straight x to the power of 4 minus open parentheses straight x squared plus 14 straight x plus 49 close parentheses d straight x end cell row blank equals cell straight pi integral subscript negative 2 end subscript superscript 1 81 minus 18 straight x squared plus straight x to the power of 4 minus straight x squared minus 14 straight x minus 49 space d straight x end cell row blank equals cell straight pi integral subscript negative 2 end subscript superscript 1 32 minus 19 straight x squared plus straight x to the power of 4 minus 14 straight x space d straight x end cell row blank equals cell straight pi times open square brackets integral subscript negative 2 end subscript superscript 1 32 space d straight x minus integral subscript negative 2 end subscript superscript 1 19 straight x squared space d straight x plus integral subscript negative 2 end subscript superscript 1 straight x to the power of 4 space d straight x minus integral subscript negative 2 end subscript superscript 1 14 straight x space d straight x close square brackets end cell row blank equals cell straight pi times open square brackets right enclose 32 straight x minus fraction numerator 19 straight x cubed over denominator 3 end fraction plus straight x to the power of 5 over 5 minus 7 straight x squared end enclose subscript negative 2 end subscript superscript 1 close square brackets end cell end table

Maka,

begin mathsize 10px style table attributes columnalign right center left columnspacing 0px end attributes row straight V equals cell straight pi times open square brackets open parentheses 32 open parentheses 1 close parentheses minus fraction numerator 19 open parentheses 1 close parentheses cubed over denominator 3 end fraction plus open parentheses 1 close parentheses to the power of 5 over 5 minus 7 open parentheses 1 close parentheses squared close parentheses minus open parentheses 32 open parentheses negative 2 close parentheses minus fraction numerator 19 open parentheses negative 2 close parentheses cubed over denominator 3 end fraction plus open parentheses negative 2 close parentheses to the power of 5 over 5 minus 7 open parentheses negative 2 close parentheses squared close parentheses close square brackets end cell row blank equals cell straight pi open square brackets open parentheses 32 minus 19 over 3 plus 1 fifth minus 7 close parentheses minus open parentheses open parentheses negative 64 close parentheses minus fraction numerator open parentheses negative 152 close parentheses over denominator 3 end fraction plus fraction numerator open parentheses negative 32 close parentheses over denominator 5 end fraction minus 7 open parentheses 4 close parentheses close parentheses close square brackets end cell row blank equals cell straight pi open square brackets open parentheses 480 over 15 minus 95 over 15 plus 3 over 15 minus 105 over 15 close parentheses minus open parentheses negative 960 over 15 plus 760 over 15 minus 96 over 15 minus 420 over 15 close parentheses close square brackets end cell row blank equals cell straight pi open square brackets 283 over 15 minus open parentheses negative 716 over 15 close parentheses close square brackets end cell row blank equals cell straight pi open square brackets 999 over 15 close square brackets end cell row blank equals cell 66 3 over 5 straight pi end cell end table end style

Volume benda putar yang terbentuk karena daerah dibatasi y equals 9 minus x squared dan y equals x plus 7 diputar 360 degree mengelilingi sumbu x adalah 66 3 over 5 straight pi.

Jadi, jawaban yang tepat adalah E.

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