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:
      • [latex]g(x) = \frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2}[/latex]
  • Riemann Sum Approximation:
    • Left-endpoint sum:
      • [latex]L_n = \sum_{i=1}^n f(\frac{i-1}{n}) \cdot \frac{1}{n}[/latex]
    • Right-endpoint sum:
      • [latex]R_n = \sum_{i=1}^n f(\frac{i}{n}) \cdot \frac{1}{n}[/latex]
  • Bounds:
    • For decreasing functions:
      • [latex]R_n \leq \text{True Value} \leq L_n[/latex]
    • For increasing functions:
      • [latex]L_n \leq \text{True Value} \leq R_n[/latex]
  • Error Estimation:
    • Maximum error:
      • [latex]\varepsilon = \frac{\text{Upper Bound} - \text{Lower Bound}}{2}[/latex]

Consider the definite integral:

[latex]\int_{-1}^0 \sqrt{1-x^4} dx[/latex]

  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 [latex]10[/latex] 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?