The Fundamental Theorem of Calculus: Why is the Fundamental Theorem of Calculus so fundamental? What does it allow us to calculate, and what concepts does it relate? Give an overview outline of the proof of this important theorem

The Fundamental Theorem of Calculus

For definite and indefinite integrals, consider the area accumulation function.

The second fundamental theorem of calculus teaches everyone how to calculate problems involving integrals or differentiation by using a relationship between the integral and the differentiation.

• If $f$is continuous on the interval $[a,b]$, and$F\left(x\right)={\int }_a^{b}f\left(x\right)dx$is continuous on$[a,b]$and differentiable on $(a,b),$then the function $F$is anti derivative of $f$, that is, $F\text{'}\left(x\right)=f\left(x\right)$.

The second fundamental theorem of calculus is known as this.

$F\left(x\right)+C=\int f\left(x\right)dx$

• If $f$is continuous, and $F\left(X\right)+C=\int f\left(x\right)$is continuous and differentiable if $C$is a constant.$F\text{'}\left(x\right)=f\text{'}\left(x\right)$The idea that was employed was:

$$\begin{gathered} \frac{dF\left(x\right)}{dx}=f\left(x\right) \\ \to \int f\left(x\right)dx=F\left(x\right)+Cf\left(x\right)dx \\ \to {\int }_a^{b}f\left(x\right)dx=F\left(x\right) \end{gathered}$$

• This is a list of the fundamental principles that mathematicians must understand in order to answer problems involving integrals.

• The second fundamental theorem of calculus is proven.

localid="1661255289852" $\begin{array}{rcl}\frac{d{\int }_a^{b}f\left(x\right)dx}{dx}& =& \frac{d{\left|F\left(x\right)\right|}_a^b}{dx}\\ & =& \frac{d\left|F\left(b\right)-F\left(a\right)\right|}{dx}\\ & =& F\text{'}\left(x\right)\\ & =& f\text{'}\left(x\right)\end{array}$