Pertanyaan

Dengan menggunakan limit jumlah Riemann, luas daerah yang dibatasi oleh kurva f ( x ) = x 2 + 1 dan sumbu x pada interval − 1 ≤ x ≤ 3 adalah .…

Dengan menggunakan limit jumlah Riemann, luas daerah yang dibatasi oleh kurva $f (x) = x^{2} + 1$ dan sumbu x pada interval $- 1 \leq x \leq 3$ adalah .…

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I. Roy

Master Teacher

Mahasiswa/Alumni Universitas Negeri Surabaya

Jawaban terverifikasi

Pembahasan

$increment x equals fraction numerator b minus a over denominator n end fraction equals fraction numerator 3 plus 1 over denominator n end fraction equals 4 over n x subscript 0 equals negative 1 x subscript i equals x subscript 0 plus i. increment x equals negative 1 plus i.4 over n equals negative 1 plus 4 over n space i f left parenthesis x right parenthesis equals x squared plus 1 space m a k a f open parentheses x subscript i close parentheses equals x subscript i squared plus 1 equals open parentheses negative 1 plus 4 over n space i close parentheses squared plus 1 space space space space space space space equals 1 minus 8 over n space i plus 16 over n squared space i squared plus 1 space space space space space space space equals 16 over n squared space i squared minus 8 over n space i plus 2$

$table attributes columnalign right center left columnspacing 0px end attributes row blank blank cell sum from i equals 1 to n of f open parentheses x subscript i close parentheses space increment x end cell row blank equals cell sum from i equals 1 to n of open parentheses 16 over n squared space i squared minus 8 over n space i plus 2 close parentheses 4 over n end cell row blank equals cell 4 over n sum from i equals 1 to n of open parentheses 16 over n squared space i squared minus 8 over n space i plus 2 close parentheses end cell row blank equals cell 4 over n open parentheses sum from i equals 1 to n of 16 over n squared space i squared minus sum from i equals 1 to n of 8 over n space i plus sum from i equals 1 to n of 2 close parentheses end cell row blank equals cell 4 over n sum from i equals 1 to n of 16 over n squared space i squared space minus space 4 over n sum from i equals 1 to n of 8 over n space i space plus space 4 over n sum from i equals 1 to n of 2 end cell row blank equals cell 64 over n cubed sum from i equals 1 to n of space i squared minus space 32 over n squared sum from i equals 1 to n of space i space plus space 8 over n sum from i equals 1 to n of 1 end cell row blank equals cell open parentheses 64 over n cubed close parentheses open parentheses fraction numerator n left parenthesis n plus 1 right parenthesis left parenthesis 2 n plus 1 right parenthesis over denominator 6 end fraction close parentheses end cell row blank blank cell negative space open parentheses 32 over n squared close parentheses open parentheses fraction numerator n left parenthesis n plus 1 right parenthesis over denominator 2 end fraction close parentheses space plus space open parentheses 8 over n close parentheses n end cell row blank equals cell fraction numerator 32 open parentheses 2 n squared plus 3 n plus 1 close parentheses over denominator 3 n squared end fraction minus fraction numerator 16 left parenthesis n plus 1 right parenthesis over denominator n end fraction plus 8 end cell row blank equals cell 64 over 3 plus 32 over n plus fraction numerator 32 over denominator 3 n squared end fraction minus 16 minus 16 over n plus 8 end cell row blank equals cell 40 over 3 plus 16 over n plus fraction numerator 32 over denominator 3 n squared end fraction end cell row L equals cell limit as n rightwards arrow infinity of open parentheses 40 over 3 plus 16 over n plus fraction numerator 32 over denominator 3 n squared end fraction close parentheses equals 40 over 3 end cell end table$