91影视

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

$$\begin{gathered} \text{The definite integral of a continuous function}f\left(x\right);A={鈭�}_a^b鈥�f\left(x\right)dx\text{estimates the area between the graph of} \\ \text{the function}f\left(x\right)\text{and}\mathrm{x}\text{-axis on an interval}[a,b]. \end{gathered}$$

$$\begin{gathered} \text{The given functions}f\left(x\right)\text{and}g\left(x\right)\text{are continuous on interval}[a,b]\text{. Suppose the graphs of} \\ \text{Functions}f\left(x\right)\text{and}g\left(x\right)\text{are as shown in figure here.} \end{gathered}$$

$$\begin{gathered} \text{The area between the curve}f\left(x\right)\text{and}x\text{-axis, i.e., area}ABCEFA={鈭�}_a^b鈥�f\left(x\right)dx \\ \text{The area between the curve}g\left(x\right)\text{and}x\text{-axis, i.e., area}ADCEFA={鈭�}_a^b鈥�g\left(x\right)dx \\ \text{Therefore, the area between the curves}f\left(x\right)\text{and}g\left(x\right)\text{on interval}[a,b]\text{is} \\ \text{Area}ABCDA=\text{area}ABCEFA\text{- area}ADCEFA \\ ABCDA={鈭�}_a^b鈥�f\left(x\right)dx鈭�{鈭�}_a^b鈥�g\left(x\right)dx={鈭�}_a^b鈥�\left[f\right(x)鈭�g(x\left)\right]dx \\ \text{Since, the area between graphs of the functions andx-axis can have positive and negative values, however} \\ \text{negative area in actual calculations has no meaning or sometimes may be absurd, e.g., if area of square is} \\ 鈭�49\text{sq.cm then its side will be}\sqrt{鈭�49}=卤7i\mathrm{cm}\text{an imaginary value. Therefore, to avoid such an absurdity the} \\ \text{computed are is taken}A=\left|{鈭�}_b^c鈥�f\left(x\right)dx\right|\text{.} \\ \text{Hence, the area between curves}f\left(x\right)\text{and}g\left(x\right)\text{on interval}[a,b]\text{is}A=\left|{鈭�}_a^b鈥�\left[f\right(x)鈭�g(x\left)\right]dx\right|. \end{gathered}$$

$\text{The area between curves}f\left(x\right)\text{and}g\left(x\right)\text{on interval}[a,b]\text{is}\left|{鈭�}_a^b鈥�\left[f\right(x)鈭�g(x\left)\right]dx\right|\text{.}$