Contoh Soal 5 Materi Integral Tentu Fungsi Aljabar

 $\begin{array}{l}\\ 21.& \text{Pada gambar berikut, inetgral yang}\\ & \text{menyatakan luas daerah yang diarsir adalah}....\\ & \begin{array}{l}\text{a}.{\displaystyle {\int }_0^3(2x^2-8x+6)dx}\\ \text{b}.{\displaystyle {\int }_0^1(x^2-4x+3)dx-{\int }_1^3(x^2-4x+3)dx}\\ \text{c}.{\displaystyle {\int }_0^1(x^2+4x+3)dx+{\int }_1^3(x^2+4x+3)dx}\\ \text{d}.{\displaystyle {\int }_0^1(2x^2-8x+6)dx-{\int }_1^3(2x^2-8x+6)dx}\\ \text{e}.{\displaystyle {\int }_0^1(2x^2+8x+6)dx-{\int }_1^3(2x^2+8x+6)dx}\end{array}\\ \end{array}$.

$.\begin{array}{rl}& \mathbf{\text{Jawab}}:\\ & \text{Kita tentukan poin batas-batasnya, yaitu}:\\ & y=a{x}^2+bx+c=d(x-1)(x-3)\\ & \text{karena kurva melalui}(0,6),\text{maka}\\ & 6=d(0-1)(0-3)\Rightarrow d=2\\ & \text{Sehingga fungsi kurvanya adalah}:\\ & y=f(x)=2(x-1)(x-3)\\ & \iff y=2(x^2-4x+3)\\ & \iff y=2x^2-8x+6\\ & \text{Selanjutnya penentuan daerah arsir}\\ & \text{tinggal menyesuaikan dengan gambar}\end{array}$.

$\begin{array}{l}\\ 22.& \text{Jika}f\text{fungsi ganjil, maka nilai}\\ & {\displaystyle {\int }_{-2022}^{2022}f(x)dx....}\\ & \begin{array}{l}\text{a}.{\displaystyle -2022}\\ \text{b}.{\displaystyle -2{\int }_0^{2022}f(x)dx}\\ \text{c}.{\displaystyle 0}\\ \text{d}.{\displaystyle 2{\int }_0^{2022}f(x)dx}\\ \text{e}.{\displaystyle 2022}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}& \text{Perlu diketahui bahwa}\\ & f(x)=\{\begin{array}{ll}\text{fungsi genap}& \Rightarrow f(-x)=f(x)\\ \text{fungsi ganjil}& \Rightarrow f(-x)=-f(x)\end{array}\\ & \text{Untuk}\\ & \begin{array}{|cc|}\hline f\text{fungsi genap}& {\displaystyle {\int }_{-a}^{a}f(x)dx=2{\int }_0^{a}f(x)dx}\\ f\text{fungsi ganjil}& {\displaystyle {\int }_{-a}^{a}f(x)dx=0}\\ \hline\end{array}\\ & \text{Karena}f(x)\text{fungsi ganjil, maka}\\ & {\displaystyle {\int }_{-2022}^{2022}f(x)dx=0}\end{array}\end{array}$.

$\begin{array}{l}\\ 23.& \text{Jika}f\text{fungsi genap, maka nilai}\\ & {\displaystyle {\int }_{-2022}^{2022}f(x)dx....}\\ & \begin{array}{l}\text{a}.{\displaystyle -2022}\\ \text{b}.{\displaystyle -2{\int }_0^{2022}f(x)dx}\\ \text{c}.{\displaystyle 0}\\ \text{d}.{\displaystyle 2{\int }_0^{2022}f(x)dx}\\ \text{e}.{\displaystyle 2022}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}& \text{Perlu diketahui bahwa}\\ & f(x)=\{\begin{array}{ll}\text{fungsi genap}& \Rightarrow f(-x)=f(x)\\ \text{fungsi ganjil}& \Rightarrow f(-x)=-f(x)\end{array}\\ & \text{Untuk}\\ & \begin{array}{|cc|}\hline f\text{fungsi genap}& {\displaystyle {\int }_{-a}^{a}f(x)dx=2{\int }_0^{a}f(x)dx}\\ f\text{fungsi ganjil}& {\displaystyle {\int }_{-a}^{a}f(x)dx=0}\\ \hline\end{array}\\ & \text{Karena}f(x)\text{fungsi genap, maka}\\ & {\displaystyle {\int }_{-2022}^{2022}f(x)dx=2{\displaystyle {\int }_0^{2022}f(x)dx}}\end{array}\end{array}$.

 $\begin{array}{l}\\ 24.& \text{Nilai dari}{\displaystyle {\int }_{-2022}^{2022}\left|x\right|dx=....}\\ & \begin{array}{l}\text{a}.{\displaystyle -2022}\\ \text{b}.{\displaystyle -2{\int }_0^{2022}\left|x\right|dx}\\ \text{c}.{\displaystyle 0}\\ \text{d}.{\displaystyle 2{\int }_0^{2022}\left|x\right|dx}\\ \text{e}.{\displaystyle 2022}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}& \text{Karena}f(x)=\left|x\right|,\text{adalah fungsi genap},\\ & \text{maka}\\ & {\displaystyle {\int }_{-2022}^{2022}\left|x\right|dx={\displaystyle 2{\int }_0^{2022}\left|x\right|dx={2022}^2}}\end{array}\end{array}$.

$\begin{array}{l}\\ 25.& \text{Nilai dari}{\displaystyle {\int }_{-2022}^{2022}x\left|x\right|dx=....}\\ & \begin{array}{l}\text{a}.{\displaystyle \frac{-{2022}^3}{3}}\\ \text{b}.{\displaystyle -2{\int }_0^{2022}x\left|x\right|dx}\\ \text{c}.{\displaystyle 0}\\ \text{d}.{\displaystyle 2{\int }_0^{2022}x\left|x\right|dx}\\ \text{e}.{\displaystyle \frac{{2022}^3}{3}}\end{array}\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}& \text{Karena}f(x)=x\left|x\right|,\text{adalah fungsi ganjil},\\ & \text{maka}\\ & {\displaystyle {\int }_{-2022}^{2022}x\left|x\right|dx=0}\end{array}\end{array}$