# Fundamental Theorems of Calculus for Definite Integrals

### Fundamental Theorems of Calculus for Definite Integrals:

#### Fundamental Theorem of Calculus - Part I:

1. Statement:

• If $f\left(x\right)$ is continuous on $\left[a,b\right]$, and $F\left(x\right)$ is an antiderivative of $f\left(x\right)$, then: ${\int }_{a}^{b}f\left(x\right)\text{\hspace{0.17em}}dx=F\left(b\right)-F\left(a\right)$
• Here, ${\int }_{a}^{b}f\left(x\right)\text{\hspace{0.17em}}dx$ represents the net signed area between $f\left(x\right)$ and the x-axis from $x=a$ to $x=b$.
2. 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:

1. Statement:

• If $f\left(x\right)$ is continuous on an interval $\left[a,b\right]$, and $F\left(x\right)$ is any antiderivative of $f\left(x\right)$, then: $\frac{d}{dx}\left({\int }_{a}^{x}f\left(t\right)\text{\hspace{0.17em}}dt\right)=f\left(x\right)$
• The derivative of the definite integral with a variable upper limit $x$ is the original function $f\left(x\right)$.
2. 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.