Question

If two functions f(x) and g(x) defined on R intersect each other at x = a & x =b then the area enclosed between these two functions will be -

• A. ${\int }_a^b\left[f\right(x)-g(x\left)\right]dx$
• B. ${\int }_a^b\left[g\right(x)-f(x\left)\right]dx$
• C. ${\int }_a^b\left[f\right(x)-g(x\left)\right]|dx$
• D. $Noneofthese$

Answer 1

The correct option is C ${\int }_a^b\left[f\right(x)-g(x\left)\right]|dx$

We have seen that the area enclosed between f(x) and X- axis between x = a &
x = b can be given by ${\int }_a^{b}f\left(x\right)dx$. Similarly the area between g(x) and X- axis between x = a &
x = b can be given by ${\int }_a^{b}g\left(x\right)dx$.
Now if $f\left(x\right)\ge g\left(x\right)\forall xϵ[a,b]$ then the area enclosed between f(x) and g(x) will be ${\int }_a^b\left[f\right(x)-g(x\left)\right]dx$.

Similarly if $g\left(x\right)\ge f\left(x\right)$ then the area enclosed between $f\left(x\right)andg\left(x\right)willbe{\int }_a^b\left[g\right(x)-f(x\left)\right]dx$.
Thus the area enclosed depends on which function is greater. Therefore to be on safe side it’s better to take the absolute value.
So, the area enclosed will be $-{\int }_a^b|\left[f\right(x)-g(x\left)\right]|dx$