Question

By integeration we can find the area under curvre of function,can we also find the area under the under the curve of function's derivative? ​

Answer 1

We can also find the area under the curve of the given function's first derivative wrt $\displaystyle\sf {x,}$ but no need of integration! Just find values of the function for the limits each, between which we need to find area, and take the magnitude of their difference.

Suppose we are given a function $\displaystyle\sf {f(x)}$ and we need to find area under $\displaystyle\sf {f'(x)}$ between $\displaystyle\sf {x=a}$ and $\displaystyle\sf {x=b.}$

And the thing goes like this, if A is the area,

$\displaystyle\sf{\longrightarrow A=\int\limits_a^bf'(x)\ dx}$

$\displaystyle\sf{\longrightarrow A=\big[f(x)\big]_a^b}$

$\displaystyle\sf {\longrightarrow\underline {\underline {A=|f(b)-f(a)|}}}$

So as it's said earlier, just find values of the function for the limits each, between which we need to find area, and take the magnitude of their difference. This requires not much integration by the way!

Answer 2

Answer:

We can also find the area under the curve of the given function's first derivative wrt $\displaystyle\sf {x,}x$, but no need of integration! Just find values of the function for the limits each, between which we need to find area, and take the magnitude of their difference.

Suppose we are given a function \displaystyle\sf {f(x)}f(x) and we need to find area under \displaystyle\sf {f'(x)}f′(x) between $\displaystyle\sf {x=a}x=a and \displaystyle\sf {x=b.}x=b.$

And the thing goes like this, if A is the area,

$\displaystyle\sf{\longrightarrow A=\int\limits_a^bf'(x)\ dx}⟶A=a∫bf′(x) dx$

$\displaystyle\sf{\longrightarrow A=\big[f(x)\big]_a^b}⟶A=[f(x)]ab$

$\displaystyle\sf {\longrightarrow\underline {\underline {A=|f(b)-f(a)|}}}⟶A=∣f(b)−f(a)∣$