Question

What are the two fundamental theorems of calculus?

Answer 1

The two fundamental theorems of calculus are as follows:

1. First fundamental theorem: It deals with the relation between the derivative and integration

If $f\left(x\right)$ is a function that is continuous over $\left[a,b\right]$ and differentiable over $\left(a,b\right)$,then the function $F\left(x\right)$ is defined as $F\left(x\right)={\int }_a^{b}f\left(t\right)dt$ , then

$F\text{'}\left(x\right)=f\left(x\right)$ over the interval $\left[a,b\right]$.

2. Second fundamental theorem: It deals with the antiderivative for an integral of a function.

If $f\left(x\right)$ is a continuous function over $\left[a,b\right]$ and if $F\left(x\right)$ is some antiderivative of $f\left(x\right)$, then ${\int }_a^{b}f\left(x\right)dx=F\left(b\right)-F\left(a\right)$