91影视

A function f plotted on the graph on [-2,2]

Given function is f(x) = 3cos(3x)

$$\begin{gathered} \mathrm{The}\mathrm{left}\mathrm{sum}\mathrm{approximation}\mathrm{for}\mathrm{area}\mathrm{of}\mathrm{this}\mathrm{region}\mathrm{with}\mathrm{n}=8\mathrm{can}\mathrm{be}\mathrm{formulated}\mathrm{as}: \\ \begin{array}{r}\mathrm{螖}x=\frac{2鈭�(鈭�2)}{8}=\frac{1}{2};x_k=鈭�2+\frac{1}{2}k;f\left(x_{k鈭�1}\right)=鈭�2+\frac{1}{2}(k鈭�1)\\ \underset{k=1}{\overset{8}{鈭�}}鈥�f\left(x_{k鈭�1}\right)\mathrm{螖}x=\\ =\frac{1}{2}\left[f\left(鈭�2+\frac{1}{2}(1鈭�1)\right)+f\left(鈭�2+\frac{1}{2}(2鈭�1)\right)+f\left(鈭�2+\frac{1}{2}(3鈭�1)\right)\right.\\ =+f\left(鈭�2+\frac{1}{2}(4鈭�1)\right)+f\left(鈭�2+\frac{1}{2}(5鈭�1)\right)+f\left(鈭�2+\frac{1}{2}(6鈭�1)\right)\\ \left.=+f\left(鈭�2+\frac{1}{2}(7鈭�1)\right)+f\left(鈭�2+\frac{1}{2}(8鈭�1)\right)\right]\\ =\frac{1}{2}\left[f(鈭�2)+f\left(\frac{鈭�3}{2}\right)+f(鈭�1)+f\left(\frac{鈭�1}{4}\right)+f\left(0\right)+f\left(\frac{1}{2}\right)+f\left(1\right)+f\left(\frac{3}{2}\right)\right]\\ =\frac{1}{2}[2.88鈭�0.63鈭�2.97+2.20+3.00+0.21鈭�2.97鈭�0.63]\\ =\frac{1}{2}\left(1.08\right)\\ =0.54\end{array} \end{gathered}$$

$$\begin{gathered} \mathrm{The}\mathrm{area}\mathrm{on}\mathrm{the}\mathrm{interval}[-2,-1.5],[-0.5,0.5]\mathrm{and}[1.5,2]\mathrm{are}\mathrm{positive}. \\ \mathrm{The}\mathrm{area}\mathrm{on}\mathrm{the}\mathrm{interval}[-1.5,0.5],[0.5,1.5]\mathrm{are}\mathrm{negative}. \\ \mathrm{Area}={鈭�}_{-2}^{-1.5}3\cos \left(3\mathrm{x}\right)+{鈭�}_{-1.5}^{-0.5}3\cos \left(3\mathrm{x}\right)+{鈭�}_{-0.5}^{0.5}3\cos \left(3\mathrm{x}\right)+{鈭�}_{0.5}^{1.5}3\cos \left(3\mathrm{x}\right)+{鈭�}_{1.5}^{2}3\cos \left(3\mathrm{x}\right) \\ \mathrm{Area}=[-0.63-2.88]+[0.21+0.63]+[0.21-0.21]+[-0.63-0.21]+[2.88+0.63] \\ \mathrm{Area}=0 \end{gathered}$$

$$\begin{gathered} \mathrm{The}\mathrm{absolute}\mathrm{function}\mathrm{is}\mathrm{given}\mathrm{below}: \\ \left|\mathrm{f}\left(\mathrm{x}\right)\right|=\left\{\begin{array}{ll}3\cos \left(3\mathrm{x}\right)& -2鈮�\mathrm{x}鈮�-1.5\\ -3\cos \left(3\mathrm{x}\right)& -1.5鈮�\mathrm{x}鈮�-0.5\\ 3\cos \left(3\mathrm{x}\right)& -0.5鈮�\mathrm{x}鈮�0.5\\ -3\cos \left(3\mathrm{x}\right)& 0.5鈮�\mathrm{x}鈮�1.5\\ 3\cos \left(3\mathrm{x}\right)& 1.5鈮�\mathrm{x}鈮�2\end{array}\right. \\ \mathrm{Area}={鈭�}_{-2}^{-1.5}3\cos \left(3\mathrm{x}\right)-{鈭�}_{-1.5}^{-0.5}3\cos \left(3\mathrm{x}\right)+{鈭�}_{-0.5}^{0.5}3\cos \left(3\mathrm{x}\right)-{鈭�}_{0.5}^{1.5}3\cos \left(3\mathrm{x}\right)+{鈭�}_{1.5}^{2}3\cos \left(3\mathrm{x}\right) \\ \mathrm{Area}=[-0.63-2.88]-[0.21+0.63]+[0.21-0.21]-[-0.63-0.21]+[2.88+0.63] \\ \mathrm{Area}=0 \end{gathered}$$