Definite Integrals

Definite Integrals:

  1. Definition:

    • Represents the accumulation of quantities over an interval on a function.
    • Symbolically denoted as abf(x)dx, where a and b are the lower and upper limits of integration, respectively, and f(x) is the integrand.
  2. Geometric Interpretation:

    • Area under a curve: abf(x)dx represents the net area between the curve y=f(x) and the x-axis from x=a to x=b.
    • Negative area: When the curve lies below the x-axis, the area is counted negatively.
  3. Properties:

    a. Linearity: The integral of a sum is the sum of the integrals: ab[f(x)+g(x)]dx=abf(x)dx+abg(x)dx.

    b. Definite Integral of a Constant: abkdx=k(ba) where k is a constant.

    c. Symmetry Property: For an even function f(x) over a symmetric interval [a,a]: aaf(x)dx=20af(x)dx.

    d. Change of Variable: If u=g(x), then g(a)g(b)f(u)du=abf[g(x)]g(x)dx.

    e. Additivity: abf(x)dx+bcf(x)dx=acf(x)dx

    f. Constant Multiplication: abkf(x)dx=kabf(x)dx

    g. Reversing Limits: abf(x)dx=baf(x)dx

  4. Fundamental Theorem of Calculus:

    • Relates definite integrals to antiderivatives: abf(x)dx=F(b)F(a) where F(x) is an antiderivative of f(x).
  5. Riemann Sums:

    • Approximates definite integrals using a series of rectangles under a curve.
    • As the number of rectangles increases, the approximation approaches the true integral value.
  6. Substitution in Definite Integrals:

    • Utilizes the chain rule for integrals in the context of definite integrals.
    • Substitution changes the limits of integration when transforming variables.
  7. Applications:

    a. Area and Volume: Computes the area under curves, volumes of revolution, and surface areas.

    b. Physics: Computes quantities like work, displacement, and moment of inertia.

    c. Probability: Integrals are used in calculating probabilities in statistics and probability theory.

Importance of Definite Integrals:

  • Quantitative Analysis: Provides precise calculations for accumulated quantities over a given interval.

  • Real-world Applications: Crucial in physics, engineering, economics, and many other fields for modeling and analysis.

  • Theoretical Framework: Fundamental in understanding the relationship between functions, derivatives, and accumulation.