Contoh Soal 8 Materi Integral Tentu Fungsi Aljabar

 $\begin{array}{l}\\ 36.& \text{Jika}\left[x\right]\text{menyatakan bilangan bulat}\\ & \text{terbesar yang}\le x,\text{maka nilai dari}\\ & {\displaystyle {\int }_0^{n^2}\left[\sqrt{x}\right]dx=....}\\ & \begin{array}{l}\text{a}.{\displaystyle \frac{(n+1)n(n+1)}{6}}\\ \text{b}.{\displaystyle \frac{n(n-1)(n-2)}{6}}\\ \text{c}.{\displaystyle \frac{n(n+1)(2n+1)}{6}}\\ \text{d}.{\displaystyle \frac{n(n-1)(3n+1)}{6}}\\ \text{e}.{\displaystyle \frac{n(n-1)(4n+1)}{6}}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\end{array}$.

$.\begin{array}{rl}& {\displaystyle {\int }_0^{n^2}\left[\sqrt{x}\right]dx}\\ & ={\displaystyle {\int }_0^1\left[\sqrt{0}\right]dx+{\int }_1^{2^2}\left[\sqrt{1}\right]dx+{\int }_{2^2}^{3^2}\left[\sqrt{2^2}\right]dx}\\ & +{\displaystyle {\int }_{3^2}^{4^2}\left[\sqrt{3^2}\right]dx+\cdots +{\int }_{(n-1{)}^2}^{n^2}\left[\sqrt{(n-1{)}^2}\right]dx}\\ & ={\displaystyle {\int }_0^{1}0dx+{\int }_1^{4}1dx+{\int }_4^{9}2dx+{\int }_9^{16}3dx}\\ & +\cdots +{\int }_{(n-1{)}^2}^{n^2}(n-1)dx\\ & =0+x{|}_1^4+2x{|}_4^9+3x{|}_9^{16}+\cdots +(n-1)x{|}_{(n-1{)}^2}^{n^2}\\ & =0+(4-1)+2(9-4)+3(16-9)+\cdots +(n-1)(2n-1)\\ & =3+2.5+3.7+4.9+5.11+\cdots +(n-1)(2n-1)\\ & =3+10+21+36+55+\cdots +(n-1)(2n-1)\\ & (\text{Deretnya berupa barisan aritmetika tingkat 2})\\ & ={\displaystyle \frac{n(n-1)(4n+1)}{6}}\end{array}$.

DAFTAR PUSTAKA

1. Hutahaean. 1980. Kalkulus Diferensial dan Integral I. Jakarta: GRAMEDIA.
2. Kuntarti, Sulistiyono, Kurnianingsih, S. 2007.  Matematika SMA dan MA untuk Kelas XII Semester 1 Program IPA Standar ISI 2006. Jakarta: ESIS.
3. Nugroho, P.A., Gunarto, D. 2013. Big Bank Soal + Bahas Matematika SMA/MA Kelas 1,2,3. Jakarta: WAHYUMEDIA.
4. Sukino. 2015. Matematika Kelompok Peminatan Matematika dan Ilmu Alam. Jakarta: ERLANGGA.
5. Suparmin, S., Malau, A. 2014. Seri Kinomatika: Mainstream Matematika Dasar & Matematika IPA untuk Siswa SMA/MA Kelompok MIA. Bandung: YRAMA WIDYA.

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Calculus | Find the Integral of the Floor Function of x

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