Area as Definite Integral

Area as Definite Integral:

  1. Objective:

    • The definite integral of a function f(x) from a to b represents the area enclosed by the curve y=f(x), the x-axis, and the lines x=a and x=b.
  2. Formulation:

    • The area A under f(x) from x=a to x=b is denoted by: A=abf(x)dx
    • Here, f(x) should be continuous and non-negative over [a,b].
  3. Geometric Interpretation:

    • For f(x)0 over [a,b], the definite integral represents the accumulation of infinitely small vertical strips of width dx and height f(x).
    • The summation of these strips over [a,b] gives the total area.
  4. Area between Curve and x-axis:

    • If f(x)0 for x in [a,b], the definite integral gives the area between the curve and the x-axis over that interval.
  5. Area Enclosed by Curves:

    • When finding the area between two curves f(x) and g(x) over [a,b], the area is given by: A=abf(x)g(x)dx where f(x)g(x)represents the vertical distance between the curves.
  6. Negative Area:

    • If f(x)0 over [a,b], the integral abf(x)dx gives the negative of the area between the curve and the x-axis.
  7. Applications:

    • Crucial in various fields like physics, economics, engineering for calculating quantities such as total displacement, work done, and accumulated values over time.
  8. Visualization and Interpretation:

    • Graphical representations and visualizations aid in understanding the relationship between the definite integral and area under curves.
    • Riemann sums provide a way to approximate area and understand the process behind definite integrals.