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Tentukan luas daerah yang dibatasi oleh: f. kurva y=x3+9x, sumbu X, garis x=−1 dan x=1

Pertanyaan

Tentukan luas daerah yang dibatasi oleh:

f. kurva begin mathsize 14px style y equals x cubed plus 9 x end style, sumbu undefined, garis begin mathsize 14px style x equals negative 1 end style dan undefined 

F. Noor

Master Teacher

Jawaban terverifikasi

Jawaban

luas daerah kurva tersebut adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 19 over 2 space satuan space luas end cell end table.

Pembahasan

Ingat kembali luas daerah menggunakan konsep integral tentu:

integral subscript a superscript b f open parentheses x close parentheses space d x equals integral subscript a superscript k f open parentheses x close parentheses space d x plus integral subscript k superscript b f open parentheses x close parentheses space d x

Pertama menentukan luas integral dengan batas open square brackets negative 1 comma space 0 close square brackets:

table attributes columnalign right center left columnspacing 0px end attributes row cell Luas space 1 end cell equals cell integral subscript negative 1 end subscript superscript 0 open parentheses 0 minus y close parentheses space d x end cell row blank equals cell negative integral subscript negative 1 end subscript superscript 0 open parentheses y close parentheses space d x end cell row blank equals cell negative integral subscript negative 1 end subscript superscript 0 open parentheses x cubed plus 9 x close parentheses space d x end cell row blank equals cell negative open square brackets 1 fourth x to the power of 4 plus 9 over 2 x squared close square brackets subscript negative 1 end subscript superscript 0 end cell row blank equals cell negative open square brackets open parentheses 1 fourth open parentheses 0 close parentheses to the power of 4 plus 9 over 2 open parentheses 0 close parentheses squared close parentheses minus open parentheses 1 fourth open parentheses negative 1 close parentheses to the power of 4 plus 9 over 2 open parentheses negative 1 close parentheses squared close parentheses close square brackets end cell row blank equals cell negative open square brackets open parentheses 0 minus open parentheses 1 fourth plus 9 over 2 close parentheses close parentheses close square brackets end cell row blank equals cell 1 fourth plus 18 over 4 end cell row blank equals cell 19 over 4 end cell end table   

Kedua menentukan luas integral dengan batas open square brackets 0 comma 1 close square brackets:

table attributes columnalign right center left columnspacing 0px end attributes row cell Luas space 2 end cell equals cell integral subscript 0 superscript 1 open parentheses 0 minus y close parentheses space d x end cell row blank equals cell integral subscript 0 superscript 1 open parentheses y close parentheses space d x end cell row blank equals cell integral subscript 0 superscript 1 open parentheses x cubed plus 9 x close parentheses space d x end cell row blank equals cell open square brackets 1 fourth x to the power of 4 plus 9 over 2 x squared close square brackets subscript 0 superscript 1 end cell row blank equals cell open square brackets open parentheses 1 fourth open parentheses 1 close parentheses to the power of 4 plus 9 over 2 open parentheses 1 close parentheses squared close parentheses minus open parentheses 1 fourth open parentheses 0 close parentheses to the power of 4 plus 9 over 2 open parentheses 0 close parentheses squared close parentheses close square brackets end cell row blank equals cell open square brackets open parentheses 1 fourth plus 9 over 2 close parentheses minus 0 close square brackets end cell row blank equals cell 1 fourth plus 18 over 4 end cell row blank equals cell 19 over 4 end cell end table  

Kemudian jumlahkan luas 1 dan luas 2:

table attributes columnalign right center left columnspacing 0px end attributes row cell Luas space kurva end cell equals cell Luas space 1 plus Luas space 2 end cell row blank equals cell 19 over 4 plus 19 over 4 end cell row blank equals cell 38 over 4 end cell row blank equals cell 19 over 2 space satuan space luas end cell end table 

Dengan demikian, luas daerah kurva tersebut adalah table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell 19 over 2 space satuan space luas end cell end table.

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