Let's analyze each part of the question based on the graphs and choices provided:

1. For graph number 1:

   • This graph shows rectangular approximations with the rectangles touching the function curve on the left side of each interval, meaning it uses the left rule.
   • Since the function appears to be increasing over the interval, the left rule typically gives an underestimate because the rectangles do not fully cover the area under the curve.

   Answer:

   • Integration method: A. left rule
   • Over or underestimate: B. under

2. For graph number 2:

   • In this graph, the rectangles touch the curve at the right side of each interval, indicating the use of the right rule.
   • With an increasing function, the right rule generally gives an overestimate because the rectangles extend above the curve on each interval.

   Answer:

   • Integration method: C. right rule
   • Over or underestimate: B. over

3. For graph number 3:

   • You are asked to sketch an estimate using the right rule with $n = 2$ subdivisions.
   • To do this, divide the interval into two equal parts and create rectangles with heights determined by the right endpoint of each subinterval.
   • Whether it’s an overestimate or underestimate depends on the shape of the function. If the function is increasing over the interval, the right rule will produce an overestimate.

   Answer:

   • Over or underestimate: B. over

4. For graph number 4:

   • You need to use the trapezoid rule with $n = 2$ subdivisions.
   • This involves dividing the interval into two parts and using trapezoids to approximate the area.
   • The trapezoid rule can either overestimate or underestimate, depending on the function's concavity. If the function is concave down, as it appears to be in this graph, the trapezoid rule gives an underestimate.

   Answer:

   • Over or underestimate: A. under

Would you like a more detailed explanation for any part of this problem, or do you have further questions?

Here are 5 related questions to deepen your understanding:

1. How do you determine whether the left or right rule will result in an overestimate or underestimate?
2. What is the impact of increasing the number of subdivisions ($n$) on the accuracy of Riemann sum approximations?
3. How does the trapezoid rule differ from the midpoint rule in terms of accuracy?
4. What types of functions (increasing, decreasing, concave up, concave down) result in overestimates or underestimates for different rules?
5. Why might the trapezoid rule be more accurate than the left or right rule?

Tip: Remember that as the number of subdivisions ($n$) increases, all Riemann sum methods (left, right, midpoint, trapezoid) become more accurate and converge toward the exact integral value.