Fundamental Theorems of Calculus for Definite Integrals
Fundamental Theorems of Calculus for Definite Integrals:
Fundamental Theorem of Calculus  Part I:

Statement:
 If $f(x)$ is continuous on $[a,b]$, and $F(x)$ is an antiderivative of $f(x)$, then: ${\int}_{a}^{b}f(x)\text{\hspace{0.17em}}dx=F(b)F(a)$
 Here, ${\int}_{a}^{b}f(x)\text{\hspace{0.17em}}dx$ represents the net signed area between $f(x)$ and the xaxis from $x=a$ to $x=b$.

Interpretation:
 Relates the definite integral of a function to its antiderivative evaluated at the interval endpoints.
 Computes the net area under a curve between the specified limits.
Fundamental Theorem of Calculus  Part II:

Statement:
 If $f(x)$ is continuous on an interval $[a,b]$, and $F(x)$ is any antiderivative of $f(x)$, then: $\frac{d}{dx}({\int}_{a}^{x}f(t)\text{\hspace{0.17em}}dt)=f(x)$
 The derivative of the definite integral with a variable upper limit $x$ is the original function $f(x)$.

Interpretation:
 Links differentiation and integration, stating that differentiation undoes the operation of integration.
 Computes the rate of change of the accumulated quantity represented by the integral.
Importance of Fundamental Theorems:

Connection Between Concepts: Unifies concepts of differentiation and integration, showcasing their inverse relationship.

Practical Applications: Enables computations of areas, velocities, volumes, and other accumulated quantities.

Theoretical Foundation: Forms the basis for more advanced calculus concepts and techniques.