It looks like you're working on numerical integration using left Riemann sums (L₂), right Riemann sums (R₂), midpoint sums (M₂), and the trapezoidal rule (T₂). However, I need the graph or function to compute these values.

Could you upload the graph or provide the function and interval you're working with? That way, I can calculate the sums and determine whether they are underestimates or overestimates of the integral I.

In general:

(a) Finding L₂, R₂, and M₂:
These are approximations of the integral based on dividing the interval into 2 subintervals:

• L₂ uses left endpoints.
• R₂ uses right endpoints.
• M₂ uses midpoints.

(b) Underestimate or overestimate?

• If the function is increasing, L₂ underestimates and R₂ overestimates.
• If the function is decreasing, L₂ overestimates and R₂ underestimates.
• M₂ can be either, depending on concavity.

(c) Finding T₂ and comparison with I:
T₂ is the average of L₂ and R₂:
$T_2 = \frac{L_2 + R_2}{2}$ T₂ is usually more accurate than L₂, R₂, or M₂ alone.

(d) Ordering Lₙ, Rₙ, Mₙ, Tₙ, and I:
For an increasing function:
$L_n \leq M_n \leq T_n \leq I \leq R_n$ For a decreasing function:
$R_n \leq I \leq T_n \leq M_n \leq L_n$

Let me know if you can provide the graph or function!

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5 Related Questions:

1. How do you compute Riemann sums for a given function?
2. Why is the trapezoidal rule generally more accurate than left or right sums?
3. What happens to these approximations as n (the number of subintervals) increases?
4. How does concavity affect whether Mn is an underestimate or overestimate?
5. What is Simpson's Rule, and how does it compare to the trapezoidal rule?

Tip:

If you increase n (more subintervals), all approximations get closer to I!