Fundamental Theorems of Calculus for Definite Integrals

Fundamental Theorems of Calculus for Definite Integrals:

Fundamental Theorem of Calculus - Part I:

  1. Statement:

    • If f(x) is continuous on [a,b], and F(x) is an antiderivative of f(x), then: abf(x)dx=F(b)F(a)
    • Here, abf(x)dx represents the net signed area between f(x) 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(x) is continuous on an interval [a,b], and F(x) is any antiderivative of f(x), then: ddx(axf(t)dt)=f(x)
    • The derivative of the definite integral with a variable upper limit x is the original function f(x).
  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.