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Jika daerah yang dibatasi oleh kurva x = 3 − y 2 dan garis x - y = 1 pada daerah x positif diputar mengelilingi sumbu y, maka volume benda putar yang terbentuk adalah...

Jika daerah yang dibatasi oleh kurva  dan garis x - y = 1 pada daerah x positif diputar mengelilingi sumbu y, maka volume benda putar yang terbentuk adalah...

  1. 5 over 4 straight pi left parenthesis 3 over 17 plus 6 square root of 3 right parenthesis satuan volume

  2. 4 over 5 straight pi left parenthesis 3 over 17 plus 6 square root of 3 right parenthesis satuan volume

  3. 5 over 4 straight pi left parenthesis 17 over 3 plus 6 square root of 3 right parenthesis satuan volume

  4. 4 over 5 straight pi left parenthesis 17 over 3 plus 6 square root of 3 right parenthesis satuan volume

  5. 5 over 4 straight pi left parenthesis 3 over 17 plus 3 square root of 6 right parenthesis satuan volume

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S. Amamah

Master Teacher

Mahasiswa/Alumni Universitas Negeri Malang

Jawaban terverifikasi

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Pembahasan

Kita buat gambarnya dahulu Cari titik potong kurva dan garis Cari titik potong kurva dan sumbu-y Sehingga dapat dihitung volumenya adalah

Kita buat gambarnya dahulu

Cari titik potong kurva dan garis

x equals x 3 minus y squared equals y plus 1 y squared plus y minus 2 equals 0 left parenthesis y minus 2 right parenthesis left parenthesis y plus 1 right parenthesis equals 0 y equals 2 space a t a u space y equals negative 1

Cari titik potong kurva dan sumbu-y

x equals 0 3 minus y squared equals 0 left parenthesis square root of 3 minus y right parenthesis left parenthesis square root of 3 plus y right parenthesis equals 0 y equals square root of 3 a t a u space y equals negative square root of 3

 

Sehingga dapat dihitung volumenya adalah

begin mathsize 12px style V equals straight pi open parentheses integral subscript negative square root of 3 end subscript superscript negative 1 end superscript left parenthesis 3 minus straight y squared right parenthesis squared d straight y plus integral subscript negative 1 end subscript superscript 1 left parenthesis 3 minus straight y squared right parenthesis squared minus left parenthesis straight y plus 1 right parenthesis squared dy close parentheses  straight V equals straight pi open parentheses integral subscript negative square root of 3 end subscript superscript negative 1 end superscript left parenthesis 9 minus 6 straight y squared plus straight y to the power of 4 right parenthesis space d straight y plus integral subscript negative 1 end subscript superscript 1 left parenthesis 9 minus 6 straight y squared plus straight y to the power of 4 right parenthesis minus left parenthesis straight y squared plus 2 straight y plus 1 right parenthesis space dy close parentheses straight V equals straight pi open parentheses integral subscript negative square root of 3 end subscript superscript negative 1 end superscript left parenthesis 9 minus 6 straight y squared plus straight y to the power of 4 right parenthesis space d straight y plus integral subscript negative 1 end subscript superscript 1 left parenthesis straight y to the power of 4 minus 7 straight y squared minus 2 straight y plus 8 right parenthesis space dy close parentheses straight V equals straight pi open parentheses open parentheses right enclose 9 straight y minus 2 straight y cubed plus 1 fifth straight y to the power of 5 end enclose blank with negative square root of 3 below and negative 1 on top close parentheses plus right enclose 1 fifth straight y to the power of 5 minus 7 over 3 straight y cubed minus straight y squared plus 8 straight y end enclose blank with negative 1 below and 1 on top close parentheses straight V equals straight pi open parentheses 9 open parentheses open parentheses negative 1 close parentheses minus open parentheses negative square root of 3 close parentheses close parentheses minus 2 open parentheses open parentheses negative 1 close parentheses cubed minus left parenthesis negative square root of 3 right parenthesis cubed close parentheses plus 1 fifth open parentheses open parentheses negative 1 close parentheses to the power of 5 minus open parentheses negative square root of 3 close parentheses to the power of 5 close parentheses close parentheses space space space space space space space plus open parentheses 1 fifth left parenthesis 1 to the power of 5 minus open parentheses negative 1 close parentheses to the power of 5 right parenthesis minus 7 over 3 open parentheses 1 cubed minus left parenthesis negative 1 right parenthesis cubed close parentheses minus open parentheses 1 squared minus left parenthesis negative 1 right parenthesis squared close parentheses plus 8 left parenthesis 1 minus left parenthesis negative 1 right parenthesis right parenthesis close parentheses straight V equals straight pi open parentheses negative 9 plus 9 square root of 3 plus 2 minus 6 square root of 3 minus 1 fifth plus 9 over 5 square root of 3 plus 2 over 5 minus 14 over 3 minus 0 plus 16 close parentheses straight V equals straight pi open parentheses 68 over 15 plus 24 over 5 square root of 3 close parentheses straight V equals 4 over 5 straight pi open parentheses 17 over 3 plus 6 square root of 3 close parentheses space satuan space volume end style

 

 

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Volume benda putar yang terjadi bila daerah yang dibatasi oleh kurva y = 4 – x 2 dan garis y = x + 2 diputar mengelilingi sumbu-x sejauh 360 o adalah… (UN 2013)

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