Contoh Soal 5 Materi Integral Tentu Fungsi Aljabar

 $\begin{array}{ll}\\ 21.&\textrm{Pada gambar berikut, inetgral yang}\\ &\textrm{menyatakan luas daerah yang diarsir adalah}\: ....\\ &\begin{array}{ll}  \textrm{a}.\quad \displaystyle \int_{0}^{3}(2x^{2}-8x+6)dx\\ \textrm{b}.\quad \displaystyle \int_{0}^{1}(x^{2}-4x+3)dx-\int_{1}^{3}(x^{2}-4x+3)dx\\ \textrm{c}.\quad \displaystyle \int_{0}^{1}(x^{2}+4x+3)dx+\int_{1}^{3}(x^{2}+4x+3)dx\\ \textrm{d}.\quad \color{red}\displaystyle \int_{0}^{1}(2x^{2}-8x+6)dx-\int_{1}^{3}(2x^{2}-8x+6)dx\\ \textrm{e}.\quad \displaystyle \int_{0}^{1}(2x^{2}+8x+6)dx-\int_{1}^{3}(2x^{2}+8x+6)dx\end{array}\\\\ \end{array}$.

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

$\begin{array}{ll}\\ 22.&\textrm{Jika}\: \: f\: \: \textrm{fungsi ganjil, maka nilai}\\ &\displaystyle \int_{-2022}^{2022}f(x)\: dx\: ....\\ &\begin{array}{ll}  \textrm{a}.\quad \displaystyle -2022\\ \textrm{b}.\quad \displaystyle -2\int_{0}^{2022}f(x)dx\\ \textrm{c}.\quad \color{red}\displaystyle 0\\ \textrm{d}.\quad \displaystyle 2\int_{0}^{2022}f(x)dx\\ \textrm{e}.\quad \displaystyle 2022\end{array}\\\\ &\textbf{Jawab}:\\ &\begin{aligned}&\textrm{Perlu diketahui bahwa}\\ &f(x)=\begin{cases} \textrm{fungsi genap} &\Rightarrow f(-x)=f(x) \\  \textrm{fungsi ganjil} &\Rightarrow f(-x)=-f(x)  \end{cases}\\ &\textrm{Untuk}\\ &\begin{array}{|c|c|}\hline f\: \textrm{fungsi genap}&\displaystyle \int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx\\\hline f\: \textrm{fungsi ganjil}&\displaystyle \int_{-a}^{a}f(x)dx=0\qquad\qquad\: \\\hline \end{array}\\ &\textrm{Karena}\: \: f(x)\: \: \textrm{fungsi ganjil, maka}\\ &\color{red}\displaystyle \int_{-2022}^{2022}f(x)\: dx=0 \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 23.&\textrm{Jika}\: \: f\: \: \textrm{fungsi genap, maka nilai}\\ &\displaystyle \int_{-2022}^{2022}f(x)\: dx\: ....\\ &\begin{array}{ll}  \textrm{a}.\quad \displaystyle -2022\\ \textrm{b}.\quad \displaystyle -2\int_{0}^{2022}f(x)dx\\ \textrm{c}.\quad \displaystyle 0\\ \textrm{d}.\quad \color{red}\displaystyle 2\int_{0}^{2022}f(x)dx\\ \textrm{e}.\quad \displaystyle 2022\end{array}\\\\ &\textbf{Jawab}:\\ &\begin{aligned}&\textrm{Perlu diketahui bahwa}\\ &f(x)=\begin{cases} \textrm{fungsi genap} &\Rightarrow f(-x)=f(x) \\  \textrm{fungsi ganjil} &\Rightarrow f(-x)=-f(x)  \end{cases}\\ &\textrm{Untuk}\\ &\begin{array}{|c|c|}\hline f\: \textrm{fungsi genap}&\displaystyle \int_{-a}^{a}f(x)dx=2\int_{0}^{a}f(x)dx\\\hline f\: \textrm{fungsi ganjil}&\displaystyle \int_{-a}^{a}f(x)dx=0\qquad\qquad\: \\\hline \end{array}\\ &\textrm{Karena}\: \: f(x)\: \: \textrm{fungsi genap, maka}\\ &\color{red}\displaystyle \int_{-2022}^{2022}f(x)\: dx=2\displaystyle \int_{0}^{2022}f(x)dx \end{aligned} \end{array}$.

 $\begin{array}{ll}\\ 24.&\textrm{Nilai dari}\: \: \displaystyle \int_{-2022}^{2022}\left | x \right |\: dx\: =....\\ &\begin{array}{ll}  \textrm{a}.\quad \displaystyle -2022\\ \textrm{b}.\quad \displaystyle -2\int_{0}^{2022}\left | x \right |dx\\ \textrm{c}.\quad \displaystyle 0\\ \textrm{d}.\quad \color{red}\displaystyle 2\int_{0}^{2022}\left | x \right |dx\\ \textrm{e}.\quad \displaystyle 2022\end{array}\\\\ &\textbf{Jawab}:\\ &\begin{aligned}&\textrm{Karena}\: \: f(x)=\left | x \right |,\: \: \textrm{adalah fungsi genap},\\ & \textrm{maka}\\ &\displaystyle \int_{-2022}^{2022}\left | x \right |\: dx\: =\color{red}\displaystyle 2\int_{0}^{2022}\left | x \right |dx=2022^{2} \end{aligned} \end{array}$.

$\begin{array}{ll}\\ 25.&\textrm{Nilai dari}\: \: \displaystyle \int_{-2022}^{2022}x\left | x \right |\: dx\: =....\\ &\begin{array}{ll}  \textrm{a}.\quad \displaystyle\frac{ -2022^{3}}{3}\\ \textrm{b}.\quad \displaystyle -2\int_{0}^{2022}x\left | x \right |dx\\ \textrm{c}.\quad \color{red}\displaystyle 0\\ \textrm{d}.\quad \displaystyle 2\int_{0}^{2022}x\left | x \right |dx\\ \textrm{e}.\quad \displaystyle \frac{2022^{3}}{3}\end{array}\\\\ &\textbf{Jawab}:\\ &\begin{aligned}&\textrm{Karena}\: \: f(x)=x\left | x \right |,\: \: \textrm{adalah fungsi ganjil},\\ & \textrm{maka}\\ &\displaystyle \int_{-2022}^{2022}x\left | x \right |\: dx\: =\color{red}0 \end{aligned} \end{array}$