Geometrical Interpretation of Definite Integral

Geometrical Interpretation of Definite Integrals:

  1. Area Under a Curve:

    • The definite integral represents the accumulated net area between a function and the x-axis over a specified interval [a,b].
    • The integral abf(x)dx computes the net area of the region bounded by f(x), the x-axis, and the vertical lines x=a and x=b.
  2. Partitioning the Interval:

    • To estimate area, partition the interval [a,b] into n subintervals.
    • Width of each subinterval: Δx=ban.
  3. Riemann Sum:

    • Approximates the area using rectangles whose heights are determined by the function values at specific points within each subinterval.
    • The sum of the areas of these rectangles approaches the area under the curve as n approaches infinity.
  4. Definite Integral as Limit of Riemann Sums:

    • As n becomes infinitely large and Δx tends to zero, the Riemann sum approaches the definite integral: abf(x)dx=limni=1nf(xi)Δx.
  5. Positive and Negative Areas:

    • Positive values of f(x) contribute to positive areas above the x-axis.
    • Negative values of f(x) contribute to negative areas below the x-axis.
  6. Interpretation with Functions:

    a. Above and Below the x-axis: For f(x)0 over [a,b], the definite integral represents the area above the x-axis.

    b. Negative Function Values: When f(x)0 over [a,b], the integral represents the negative of the area below the x-axis.

Importance of Geometric Interpretation:

  • Visualization of Integrals: Provides a visual understanding of the concept of definite integrals as areas under curves.

  • Applications in Geometry: Facilitates computations of areas of irregular shapes and volumes of solids by using integrals.

  • Understanding Function Behavior: Helps in understanding the behavior of functions over specific intervals through their graphical representations.