Hitunglah volume benda putar yang dibatasi oleh y=2+x2 dan y=10−x2 yang diputar terhadap sumbu x!

Pertanyaan

Hitunglah volume benda putar yang dibatasi oleh begin mathsize 14px style y equals 2 plus x squared end style dan begin mathsize 14px style y equals 10 minus x squared end style yang diputar terhadap sumbu begin mathsize 14px style x end style!

N. Puspita

Master Teacher

Jawaban terverifikasi

Jawaban

volume benda putar yang terbentuk adalah begin mathsize 14px style 256 straight pi end style.

Pembahasan

Pembahasan

Akan ditentukan titik potong begin mathsize 14px style y equals 2 plus x squared space dan space y equals 10 minus x squared end style.
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row cell 2 plus x squared end cell equals cell 10 minus x squared end cell row cell 2 x squared minus 8 end cell equals 0 row cell x squared minus 4 end cell equals 0 row cell left parenthesis x minus 2 right parenthesis left parenthesis x plus 2 right parenthesis end cell equals 0 row blank blank cell x subscript 1 equals 2 comma space x subscript 2 equals negative 2 end cell end table end style
 

didapat batas integralnya begin mathsize 14px style 2 space dan space minus 2 end style.

Gambar daerah kurvanya seperti berikut.
 


 

Volume benda putarnya.
 

begin mathsize 14px style table attributes columnalign right center left columnspacing 0px end attributes row V equals cell straight pi integral subscript negative 2 end subscript superscript 2 open parentheses 10 minus x squared close parentheses squared minus open parentheses 2 plus x squared close parentheses squared d x end cell row blank equals cell straight pi integral subscript negative 2 end subscript superscript 2 left parenthesis 100 minus 20 straight x squared plus straight x to the power of 4 right parenthesis minus left parenthesis 4 minus 4 straight x squared plus straight x to the power of 4 right parenthesis d straight x end cell row blank equals cell straight pi integral subscript negative 2 end subscript superscript 2 96 minus 24 straight x squared d straight x end cell row blank equals cell straight pi open square brackets 96 straight x minus 8 straight x cubed close square brackets subscript negative 2 end subscript superscript 2 end cell row blank equals cell straight pi open square brackets left parenthesis 96 times 2 minus 8.2 cubed right parenthesis minus left parenthesis 96 times left parenthesis negative 2 right parenthesis minus 8. left parenthesis negative 2 right parenthesis cubed right parenthesis close square brackets end cell row blank equals cell straight pi open square brackets left parenthesis 192 minus 64 right parenthesis minus left parenthesis negative 192 plus 64 right parenthesis close square brackets end cell row blank equals cell straight pi open square brackets 128 minus left parenthesis negative 128 right parenthesis close square brackets end cell row blank equals cell 256 straight pi end cell end table end style
 

Jadi, volume benda putar yang terbentuk adalah begin mathsize 14px style 256 straight pi end style.

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