Overestimate or Underestimate? Riemann Sum Error
Tell whether a left, right, midpoint or trapezoidal sum over- or underestimates using increasing/decreasing and concavity, then see how error shrinks.
Is a Riemann Sum an Overestimate or Underestimate?
You slice a curve into rectangles and add them up. The calculator gives a number, but is that number too high or too low? The question of whether a riemann sum overestimate or underestimate the true area comes down to two things about the function: whether it is increasing or decreasing, and whether it bends up or down. The rules, justification sentences for an exam, and failure cases when the function changes direction in the middle are provided.
Left and Right Riemann Sums: Increasing vs. Decreasing f
For a left riemann sum, you use the left endpoint of each subinterval to set the rectangle height. For a right riemann sum, you use the right endpoint. The monotonicity of the function determines which one is an overestimate and which is an underestimate.
If f is Increasing
A left riemann sum uses heights from the left side of each subinterval, where the function is lower. The rectangles fall short of the curve. Left sum is an underestimate. A right sum uses heights from the right side, where the function is higher. The rectangles overshoot. Right sum is an overestimate.
If f is Decreasing
Now the left endpoint gives a higher height than the curve at the right side of the subinterval. Left sum is an overestimate. The right endpoint gives a lower height. Right sum is an underestimate.
Sketch: draw an increasing curve from left to right. Left rectangles sit below the curve; right rectangles stick above it. For a decreasing curve, the picture reverses.
Midpoint and Trapezoid: Concave Up vs. Concave Down
Midpoint and trapezoidal rule behavior depends on concavity, not monotonicity. The midpoint rectangle uses the height at the middle of the subinterval. The trapezoid uses the average of the two endpoint heights. The second derivative tells you which way the error goes.
If f is Concave Up (f'' > 0)
The curve sits above its secant line on each subinterval. The trapezoid uses that secant line, so its area falls below the curve. Trapezoidal rule is an underestimate. The midpoint rectangle, however, uses a height at the center, which for a concave-up curve sits above the curve's height at the center because the curve is above its chord. The rectangle overshoots. Midpoint rule is an overestimate.
If f is Concave Down (f'' < 0)
The curve sits below its secant line. Trapezoidal rule is an overestimate. The midpoint rectangle now falls short. Midpoint rule is an underestimate.
Sketch: draw a concave-up curve. The straight line connecting the endpoints lies below the curve; the midpoint of that chord lies below the curve as well, but the rectangle height at the midpoint of the subinterval is above the curve because the curve is higher at the midpoint than the chord. This is the standard textbook picture from Stewart, Calculus, Section 7.7.
| Function Behavior | Left Sum | Right Sum | Midpoint Rule | Trapezoidal Rule |
|---|---|---|---|---|
| Increasing | Underestimate | Overestimate | Depends on concavity | Depends on concavity |
| Decreasing | Overestimate | Underestimate | Depends on concavity | Depends on concavity |
| Concave up (f'' > 0) | Depends on monotonicity | Depends on monotonicity | Overestimate | Underestimate |
| Concave down (f'' < 0) | Depends on monotonicity | Depends on monotonicity | Underestimate | Overestimate |
What To Do When f Changes Direction on the Interval
A function that increases then decreases inside [a,b] breaks the simple rules. You cannot claim an overall overestimate or underestimate for the whole sum based on monotonicity or concavity alone. The AP Calculus CED Topic 6.2 justification language says monotonicity must hold over the entire interval for the claim to be valid.
Failure case: the function f(x)=x²-4x on [0,4] decreases on [0,2] and increases on [2,4]. A left sum overestimates on the decreasing part and underestimates on the increasing part. The total error could be positive, negative, or near zero depending on the partition. The only safe statement is: the exact value lies between L_n and R_n if the function is monotonic. For non-monotonic functions, you cannot guarantee that either.
What to do: split the interval at the critical point where the derivative changes sign. Apply the rule on each subinterval separately, then compare the partial sums if you need a bound. In practice, for a table of values you can only state the direction of the error for each subinterval where the function is monotonic.
Absolute and Relative Error, and How They Fall as n Doubles
The absolute error is |exact - approximation|. The relative error is (absolute error / |exact|) × 100%. Both are what the calculator reports, but the meaningful number for exams is the absolute error bound.
Error Bound Formulas from Stewart, Section 7.7
Stewart gives error bounds for the midpoint rule (M_n), trapezoidal rule (T_n), and Simpson's rule (S_n). Let K be the maximum of |f''(x)| on [a,b] for M_n and T_n, and the maximum of |f^(4)(x)| for S_n. Then:
|E_M| ≤ K(b-a)^3 / (24n^2)
|E_T| ≤ K(b-a)^3 / (12n^2)
|E_S| ≤ K(b-a)^5 / (180n^4)
These are worst-case guarantees, not actual errors. Doubling n divides the bound for M_n and T_n by a factor of 4, because n appears squared in the denominator. For Simpson's rule, doubling n divides the bound by 16, because n^4 is in the denominator. This is why increasing n is the first thing to try when the relative error exceeds 5%.
Absolute and Relative Error in Practice
If you have the exact integral from an antiderivative, compute the absolute error directly. If not, the error bound is your only tool. A relative error below 0.1% means the approximation is good enough for almost any application. A relative error above 10% means you should increase n significantly or switch to Simpson's rule if the function is smooth.
Justification Sentences for an Exam Answer
The AP Calculus CED Topic 6.2 specifies exact language for justifying whether a Riemann sum is an overestimate or underestimate. Use these sentences verbatim.
For Left and Right Sums
"Since f is increasing, L_n is an underestimate and R_n is an overestimate."
"Since f is decreasing, L_n is an overestimate and R_n is an underestimate."
For Midpoint and Trapezoidal Rules
"Since f is concave up, M_n is an overestimate and T_n is an underestimate."
"Since f is concave down, M_n is an underestimate and T_n is an overestimate."
When You Have Both Monotonicity and Concavity
The exam sometimes gives a combined ordering. For an increasing, concave-up function: L_n < T_n < exact < M_n < R_n. For an increasing, concave-down function: L_n < M_n < exact < T_n < R_n. For a decreasing, concave-up function: R_n < M_n < exact < T_n < L_n. For a decreasing, concave-down function: R_n < T_n < exact < M_n < L_n.
Never mention concavity when the question asks only about monotonicity. The AP scoring guide awards the point for the correct reference to monotonicity alone. Adding concavity is not wrong but can cost time.
Common Questions
Can a left Riemann sum ever be an overestimate for an increasing function?
No. If the function is strictly increasing, the left sum always underestimates. The left endpoint is the lowest point in each subinterval. The rectangle height is always below the curve at the right side of the subinterval. This is a logical consequence of the definition, not a convention.
What justifies the trapezoidal rule being an underestimate when the function is concave up?
A concave-up function lies above its secant line on each subinterval. The trapezoid uses that secant line as its top. Since the curve is above the line, the trapezoid area is less than the true area under the curve. The second derivative being positive is the formal condition.
How do I know which method gives a smaller error for the same n?
For smooth functions, midpoint and trapezoidal rules usually have smaller absolute error than left or right sums. Simpson's rule is more accurate than both for functions with a continuous fourth derivative. The error bound formulas from Stewart, Section 7.7 show that midpoint and trapezoidal errors are proportional to 1/n^2, while left and right errors are only proportional to 1/n.
What if the function changes concavity on the interval?
You cannot claim a global overestimate or underestimate for the midpoint or trapezoidal rule. Split the interval at the inflection point, apply the rule on each piece, and then compare the partial sums if you need a bound. The error bound formulas still apply because they use the maximum of |f''(x)| over the whole interval, which gives a worst-case guarantee.
Does doubling n always cut the absolute error in half?
No. For left and right sums, doubling n roughly halves the error because the error is proportional to 1/n. For midpoint and trapezoidal rules, doubling n roughly quarters the error because the error is proportional to 1/n^2. This is an approximate behavior, not a guarantee. The actual error can increase slightly for a specific n before the trend takes over.
What is the difference between absolute error and relative error on a calculator?
Absolute error is the raw difference between the approximation and the true value, in the same units as the area. Relative error expresses that difference as a percentage of the true value. A large absolute error on a large integral may still be a small relative error. For AP exams, absolute error is usually sufficient; relative error is more useful for comparing accuracy across different integrals.
Why can't I use Simpson's rule with an odd number of subintervals?
Simpson's rule requires an even number of subintervals because it fits a parabola through three points: the endpoints and the midpoint of each pair of subintervals. With an odd number, one subinterval is left out. Stewart, Section 7.7 states this requirement explicitly. Using an odd n gives a result that is not a valid Simpson's rule approximation.