91影视

$\text{The functions}f\left(x\right)\text{continuous on interval}[a,b]\text{.}$

$$\begin{gathered} \text{The average}\left(\text{mean}\right)\text{of function}f\left(x\right)\text{for arguments}{x}_1,x_2,x_3鈥�x_n\text{is given by}\stackrel{炉}{f\left(x\right)}=\frac{f\left(x_1\right)+f\left(x_2\right)+f\left(x_3\right)鈥�f\left(x_n\right)}{n}.\text{The} \\ \text{integral;}A={鈭�}_a^b鈥�f\left(x\right)dx\text{, estimates the area between the graph of the function}f\left(x\right)\text{and}x\text{-axis on an interval} \\ [a,b]. \end{gathered}$$

$$\begin{gathered} \text{The graph of function}f\left(x\right)\text{is continuous on interval}[a,b]\text{. The average value of the function}f\left(x\right)\text{is} \\ \stackrel{炉}{f\left(x\right)}=\frac{f\left(x_1\right)+f\left(x_2\right)+f\left(x_3\right)鈥�f\left(x_n\right)}{n}=\frac{\underset{i=1}{\overset{n}{鈭�}}鈥�f\left(x_1\right)}{n}.................\left(1\right) \\ \text{Now, we divide length of interval}b鈭�a\text{into n parts such that}\frac{b鈭�a}{n}=\mathrm{螖}x,\mathrm{螖}x\text{is as small as we can take it.} \\ \text{Thercforc, substituting valuc of}n\text{rcsult (1), wc have.} \\ \stackrel{炉}{f\left(x\right)}=\frac{f\left(x_1\right)+f\left(x_2\right)+f\left(x_3\right)鈥�f\left(x_n\right)}{n}=\frac{\underset{i=1}{\overset{n}{鈭�}}鈥�f\left(x_k\right)}{\frac{b鈭�a}{\mathrm{螖}x}}=\frac{\underset{k=1}{\overset{n}{鈭�}}鈥�f\left(x_k\right)\mathrm{螖}x}{b鈭�a},.\text{But}f\left(x_k\right)\mathrm{螖}x\text{is the area of small rectangle made by} \\ \text{vertical length}f\left(x_k\right)\text{and width}\mathrm{鈻�}x\text{, and so}\underset{k=1}{\overset{n}{鈭�}}鈥�f\left(x_k\right)\mathrm{螖}x\text{is the sum of areas of all rectangles on the interval} \\ [a,b]\text{. That further implies}\underset{k=1}{\overset{n}{鈭�}}鈥�f\left(x_k\right)\mathrm{螖}x\text{is the area between the graph of function}f\left(x\right)\text{and}x\text{-axis on aninterval[a,b].} \\ \text{Since,}f\left(x\right)\text{is a continuous function, therefore, we have}\underset{k=1}{\overset{n}{鈭�}}鈥�f\left(x_k\right)\mathrm{螖}x=鈭�f\left(x\right)dx\text{.} \\ 鈬�\stackrel{炉}{f\left(x\right)}=\frac{\underset{i=1}{\overset{n}{鈭�}}鈥�f\left(x_k\right)\mathrm{螖}x}{b鈭�a}=\frac{1}{b鈭�a}{鈭�}_a^b鈥�f\left(x\right)dx \\ \text{Thus, the average value of function is}f(x{)}_{av}=\stackrel{炉}{f\left(x\right)}=\frac{1}{b鈭�a}{鈭�}_a^b鈥�f(x)dx\text{.} \end{gathered}$$

$\text{The average value of function}f\left(x\right)\text{on interval}[a,b]\text{is defined as}\stackrel{炉}{f\left(x\right)}=\frac{1}{b鈭�a}{鈭�}_a^b鈥�f\left(x\right)dx\text{.}$