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Tentukan luas daerah yang dibatasi oleh: a. kurva , sumbu , , dan ;

Tentukan luas daerah yang dibatasi oleh:

a. kurva begin mathsize 14px style y equals x squared minus 4 x plus 3 end style, sumbu begin mathsize 14px style x end stylebegin mathsize 14px style x equals 0 end style, dan begin mathsize 14px style x equals 4 end style;

  1. ...undefined 

  2. ...undefined 

Jawaban:

Perhatikan gambar di bawah ini:

 

Keterangan: kurva bergaris merah adalah kurva begin mathsize 14px style y equals x squared minus 4 x plus 3 end style 

Luas daerah yang dicari = begin mathsize 14px style straight L subscript straight I plus straight L subscript II plus straight L subscript III end style, dimana:

  • undefined    
  • begin mathsize 11px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight L subscript II end cell equals cell integral subscript 1 superscript 3 open parentheses 0 minus open parentheses x squared minus 4 x plus 3 close parentheses close parentheses d x end cell row blank equals cell integral subscript 1 superscript 3 open parentheses negative x squared plus 4 x minus 3 close parentheses d x end cell row blank equals cell negative 1 third x cubed plus 4 over 2 x squared minus 3 x vertical line subscript 1 superscript 3 end cell row blank equals cell negative 1 third x cubed plus 2 x squared minus 3 x vertical line subscript 1 superscript 3 end cell row blank equals cell open parentheses negative 1 third cross times 3 cubed plus 2 cross times 3 squared minus 3 cross times 3 close parentheses minus open parentheses negative 1 third cross times 1 cubed plus 2 cross times 1 squared minus 3 cross times 1 close parentheses end cell row blank equals cell open parentheses negative 9 plus 18 minus 9 close parentheses minus open parentheses negative 1 third plus 2 minus 3 close parentheses end cell row blank equals cell 0 minus open parentheses negative 4 over 3 close parentheses end cell row blank equals cell 4 over 3 end cell end table end style 
  • begin mathsize 11px style table attributes columnalign right center left columnspacing 0px end attributes row cell straight L subscript III end cell equals cell integral subscript 3 superscript 4 open parentheses open parentheses x squared minus 4 x plus 3 close parentheses minus 0 close parentheses d x end cell row blank equals cell integral subscript 3 superscript 4 left parenthesis x squared minus 4 x plus 3 right parenthesis d x end cell row blank equals cell 1 third x cubed minus 4 over 2 x squared plus 3 x vertical line subscript 3 superscript 4 end cell row blank equals cell 1 third x cubed minus 2 x squared plus 3 x vertical line subscript 3 superscript 4 end cell row blank equals cell left parenthesis 1 third cross times 4 cubed minus 2 cross times 4 squared plus 3 cross times 4 right parenthesis minus left parenthesis 1 third cross times 3 cubed minus 2 cross times 3 squared plus 3 cross times 3 right parenthesis end cell row blank equals cell open parentheses 64 over 3 minus 32 plus 12 close parentheses minus open parentheses 9 minus 18 plus 9 close parentheses end cell row blank equals cell 4 over 3 minus 0 end cell row blank equals cell 4 over 3 end cell end table end style 

Jadi, luas daerah diatas adalah begin mathsize 14px style straight L subscript straight I plus straight L subscript II plus straight L subscript III equals 4 over 3 plus 4 over 3 plus 4 over 3 equals 12 over 3 equals 4 end style satuan luas. undefined 

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