- 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)=1√2πe−12x2
- Riemann Sum Approximation:
- Left-endpoint sum:
- Ln=∑ni=1f(i−1n)⋅1n
- Right-endpoint sum:
- Rn=∑ni=1f(in)⋅1n
- Left-endpoint sum:
- Bounds:
- For decreasing functions:
- Rn≤True Value≤Ln
- For increasing functions:
- Ln≤True Value≤Rn
- For decreasing functions:
- Error Estimation:
- Maximum error:
- ε=Upper Bound−Lower Bound2
- Maximum error:
Consider the definite integral:
∫0−1√1−x4dx
- Can this integral be evaluated using the Fundamental Theorem of Calculus, Part 2?
- If not, provide an upper bound and a lower bound for the true value of the definite integral using 10 subintervals.
- Choose a single value from that interval to use as an approximation. What is the maximum error that could be associated with that approximation?