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Approximating Integrals: Fresh Take

  • Understand that while all differentiable functions can be derived in a straightforward formula, not all functions can be integrated into a simple antiderivative
  • Calculate bounds for the area calculations under curves when direct integration methods aren’t applicable
  • Explain how estimating the bounds of an integral affects the accuracy of the approximation

Approximating Integrals

The Main Idea 

  • Not all functions have antiderivatives in terms of elementary functions
  • Riemann sums can be used to approximate definite integrals
  • Upper and lower bounds provide an interval for the true value
  • Error estimation helps quantify the accuracy of approximations
  • Non-integrable Functions:
    • Some functions don’t have antiderivatives in terms of elementary functions
    • Example:
      • g(x)=12πe12x2
  • Riemann Sum Approximation:
    • Left-endpoint sum:
      • Ln=ni=1f(i1n)1n
    • Right-endpoint sum:
      • Rn=ni=1f(in)1n
  • Bounds:
    • For decreasing functions:
      • RnTrue ValueLn
    • For increasing functions:
      • LnTrue ValueRn
  • Error Estimation:
    • Maximum error:
      • ε=Upper BoundLower Bound2

Consider the definite integral:

011x4dx

  1. Can this integral be evaluated using the Fundamental Theorem of Calculus, Part 2?
  2. If not, provide an upper bound and a lower bound for the true value of the definite integral using 10 subintervals.
  3. Choose a single value from that interval to use as an approximation. What is the maximum error that could be associated with that approximation?