Q. 10 - Calculus

Question

The area between two curves f and g on [a, b] .

Step-by-Step Solution

Answer

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

Step 1. Given Information

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

Step 2. Formula used

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

Step 3. Definition/Explanation

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

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

Step 4. Conclusion

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