Definite integral as limit of a sum

Definite Integral as Limit of a Sum:

  1. Purpose of Definite Integral:

    • Represents the accumulation of quantities or area under a curve.
    • Denoted as abf(x)dx for a function f(x) over the interval [a,b].
  2. Riemann Sum Approximation:

    • Divides the interval [a,b] into n subintervals.
    • Width of each subinterval: Δx=ban.
  3. Summation over Subintervals:

    • For a partition P with n subintervals: S=i=1nf(ci)Δx is a sample point within the ith subinterval.
  4. Definite Integral as a Limit:

    • The definite integral of f(x) over [a,b] is expressed as a limit of Riemann sums: abf(x)dx=limni=1nf(ci)Δx as the number of subintervals approaches infinity (n).
  5. Interpretation and Geometric Meaning:

    • The definite integral represents the exact area under the curve f(x) from a to b on the x-axis.
    • As n increases (subintervals become infinitesimally small), the Riemann sum approaches the precise area.
  6. Types of Riemann Sums:

    • Left Riemann Sum: Uses left endpoints of subintervals.
    • Right Riemann Sum: Uses right endpoints of subintervals.
    • Midpoint Riemann Sum: Uses midpoints of subintervals.
  7. Properties and Applications:

    • Crucial in calculus for solving problems related to areas, volumes, work done, and various physical quantities.
    • Basis for understanding the Fundamental Theorem of Calculus and techniques like integration by substitution.
  8. Integration Techniques:

    • Helps in evaluating definite integrals of functions that might be challenging to integrate directly.

Example:

Consider the function f(x)=2x over the interval [0,2]. We want to find the definite integral of f(x) from 0 to 2 using the limit of a sum.

  1. Partition the Interval:

    • Divide the interval [0,2] into n subintervals.
    • Width of each subinterval: Δx=20n=2n.
  2. Riemann Sum:

    • For the ith subinterval, choose ci as the right endpoint xi=0+iΔx=2in.
    • The Riemann sum for the function f(x)=2x is: S=i=1nf(ci)Δx=i=1n2(2in)2n
  3. Definite Integral as Limit:

    • Express the definite integral of f(x)=2x from 0 to 2 as a limit of the Riemann sum: 022xdx=limni=1n2(2in)2n
  4. Simplify the Sum:

    • Simplify the sum to solve for the definite integral: S=limni=1n8in2=limn8n2n(n+1)2 S=limn4(n+1)n=limn(4+4n)=4
  5. Result:

    • Therefore, by using the definite integral as the limit of a sum, we found that: 022xdx=4