What is the general Riemann sum formula in sigma notation?

The general form is Σ_{i=1}^{n} f(x_i*) Δx, where Δx = (b − a)/n, and x_i* is a sample point in the i-th subinterval. The specific formula depends on which sample point you use: left, right, or midpoint.

How do I write a left Riemann sum formula?

Use x_i* = a + (i-1)Δx. The sum is Σ_{i=1}^{n} f(a + (i-1)Δx) Δx. This uses the left endpoint of each subinterval.

What is the right Riemann sum formula in sigma notation?

Use x_i* = a + iΔx. The right Riemann sum formula is Σ_{i=1}^{n} f(a + iΔx) Δx. The index i runs from 1 to n, giving endpoints x_1 through x_n.

What is the midpoint Riemann sum formula in sigma notation?

Use x̄_i = a + (i − 1/2)Δx. The midpoint Riemann sum formula is Σ_{i=1}^{n} f(a + (i − 1/2)Δx) Δx. The sample point is halfway between the left and right endpoints.

What does x_i = a + iΔx mean in the context of Riemann sums?

This formula defines the partition points of the interval [a,b]. For i=0, x_0 = a; for i=n, x_n = b. The left endpoint of the i-th subinterval is x_{i-1} = a + (i-1)Δx, and the right endpoint is x_i = a + iΔx.

How do I know whether a given summation is a left or right Riemann sum?

Check the argument of the function. If the argument uses a + (i-1)Δx, it is a left sum. If it uses a + iΔx, it is a right sum. If it uses a + (i − 1/2)Δx, it is a midpoint sum. The width Δx is the factor outside the function.

What is the failure mode when writing a Riemann sum in sigma notation?

The most common error is an indexing mistake: using i when you need i-1 or vice versa. For a left sum, the first sample point is at x_0 = a, which matches i=1 giving a + (1-1)Δx = a. For a right sum, the first sample point is at x_1 = a + Δx. Check the first term to catch this error.

Riemann Sum Formula: Left, Right, Midpoint in Σ Notation

The Riemann Sum Formula: What You Copy Into Your Notes

A student sits down to study numerical integration and needs the exact sigma notation for each Riemann sum type. No history, no context, just the formulas, ready to copy. The general Riemann sum formula, and then the specific left, right, and midpoint Riemann sum formulas, are all given in the sigma notation your textbook uses. Stewart, Calculus, sections 5.1-5.2 defines the sum as Σ f(x_i*) Δx, where Δx = (b − a)/n and x_i = a + iΔx. OpenStax Calculus Volume 1, section 5.1 uses the same notation. You can use any of the three sample points, left endpoint, right endpoint, or midpoint, and the formula adapts by changing x_i*.

The Riemann sum formula is always an approximation, not the exact area. The definite integral is the limit of the Riemann sum as n → ∞. If you need the exact value, you take that limit or use the Fundamental Theorem of Calculus. The Riemann sum is the finite step before that limit, and the choice of sample point determines the method.

Riemann Sum Formula Reference Table

Method	Sample Point x_i*	Sigma Notation	Common Use
Left Riemann Sum	x_{i-1} = a + (i-1)Δx	Σ_{i=1}^{n} f(a + (i-1)Δx) Δx	Underestimates increasing functions; overestimates decreasing ones
Right Riemann Sum	x_i = a + iΔx	Σ_{i=1}^{n} f(a + iΔx) Δx	Overestimates increasing functions; underestimates decreasing ones
Midpoint Riemann Sum	x̄_i = a + (i − 1/2)Δx	Σ_{i=1}^{n} f(a + (i − 1/2)Δx) Δx	Often more accurate than left/right for same n; not guaranteed over/underestimate
Trapezoidal Rule	(f(x_{i-1}) + f(x_i))/2	(Δx/2)[f(x_0) + 2f(x_1) + ... + 2f(x_{n-1}) + f(x_n)]	Averages left and right sums; exact for linear functions
Simpson's Rule	Parabola through three points	(Δx/3)[f(x_0) + 4f(x_1) + 2f(x_2) + ... + 4f(x_{n-1}) + f(x_n)]	Exact for cubics; requires even n

Left Riemann Sum Formula

The left Riemann sum formula uses the left endpoint of each subinterval for the height. With a partition a = x_0 < x_1 < ... < x_n = b and uniform subinterval width Δx = (b − a)/n, the left endpoint of the i-th subinterval is x_{i-1} = a + (i-1)Δx. The sum is:

Left Riemann sum = Σ_{i=1}^{n} f(a + (i-1)Δx) Δx

Use this when you know the function is monotonic. If the function is increasing on [a,b], the left Riemann sum underestimates the true area because each rectangle's height is taken at the left side, where the function value is smallest. If the function is decreasing, the left sum overestimates. OpenStax Calculus Volume 1, section 5.1 shows this with worked examples of left-endpoint approximations.

Right Riemann Sum Formula

The right Riemann sum formula takes the height at the right endpoint of each subinterval. Here x_i = a + iΔx. The sum becomes:

Right Riemann sum = Σ_{i=1}^{n} f(a + iΔx) Δx

For an increasing function, the right sum overestimates because the rectangle height is the function's largest value on the subinterval. For a decreasing function, the right sum underestimates. The AP Calculus CED Topic 6.2 specifically tests the ability to justify these over/underestimates using monotonicity. If a table of values gives only discrete f(x) data, use the right endpoint value for each subinterval. The AP Calculus AB FRQ problems often provide tables and ask for a right Riemann sum with correct units.

Midpoint Riemann Sum Formula

The midpoint Riemann sum formula uses the midpoint x̄_i = a + (i − 1/2)Δx. This point is exactly halfway between x_{i-1} and x_i. The sum is:

Midpoint Riemann sum = Σ_{i=1}^{n} f(a + (i − 1/2)Δx) Δx

The midpoint rule is generally more accurate than the left or right rule for the same number of subintervals because it balances the over- and underestimates within each subinterval. However, it is not guaranteed to be an over- or underestimate for a monotonic function. The error depends on the function's concavity. Stewart, section 7.7 gives the error bound for the midpoint rule as |E_M| ≤ K(b−a)^3 / (24 n^2), where K is the maximum of |f''(x)| on [a,b]. OpenStax Calculus Volume 2, section 3.6 defines the midpoint rule as M_n = Δx [f(x̄_1) + f(x̄_2) + ... + f(x̄_n)].

Left Riemann Sum Formula vs Right Riemann Sum Formula vs Midpoint Riemann Sum Formula: Comparing Sample Points

How the Sample Point Changes the Sum

The difference between the three methods is always the value of x_i*, the sample point. For left sums, x_i* = x_{i-1}. For right sums, x_i* = x_i. For midpoint sums, x_i* = x̄_i, the midpoint. The formula x_i = a + iΔx defines the partition points. The left Riemann sum formula uses index i-1; the right uses i; the midpoint uses halfway between them. When you write the sum in sigma notation, the index starts at 1 and goes to n in all three cases. The difference is only inside the function argument.

If you have a table of values at unevenly spaced x-values, you cannot use the formulas above exactly because Δx varies per subinterval. In that case, compute each rectangle's area as f(x_i*) * (x_i - x_{i-1}), then sum them. The AP Calculus BC FRQ sometimes includes non-uniform partitions, requiring this direct multiplication.

Riemann Sum Sigma Notation: Writing a Given Sum

Identify the Pattern

To write a given sum in sigma notation, identify the pattern for the sample points and the width. Suppose a problem gives you the sum Σ_{i=1}^{5} f(2 + 0.5i) * 0.5. The width Δx = 0.5, and the sample point is 2 + 0.5i, which matches x_i = a + iΔx with a = 2 and Δx = 0.5. Therefore this is a right Riemann sum over [2, 4.5] with n = 5 subintervals.

Worked Example

A car's velocity in ft/s is given at 2-second intervals: v(0)=0, v(2)=8, v(4)=12, v(6)=15, v(8)=17. Approximate the distance traveled from t=0 to t=8 using a right Riemann sum with n=4. The width Δx = (8-0)/4 = 2. The right endpoints are t=2,4,6,8. The sum is Σ_{i=1}^{4} v(2i) * 2 = (8 + 12 + 15 + 17) * 2 = 52 * 2 = 104 feet. This sum is a right Riemann sum; the sigma notation looks like Σ_{i=1}^{4} v(0 + 2i) * 2. If the velocity function is increasing, this right sum overestimates the true distance.

Upper and Lower Sums

OpenStax Calculus Volume 1, section 5.1 defines the upper sum U(f, P) = Σ M_i Δx_i and the lower sum L(f, P) = Σ m_i Δx_i, where M_i is the maximum of f on the i-th subinterval and m_i is the minimum. These are not the same as left or right sums, they depend on where the maximum and minimum occur, which might be anywhere inside the subinterval. For a monotonic function, the maximum is at one endpoint and the minimum at the other, so the upper sum equals either the left or right sum. For a general function, the upper sum is the largest possible Riemann sum for that partition, and the lower sum is the smallest. The definite integral lies between them as the partition gets finer.

Writing a Word Problem Into Sigma Notation

An AP-style problem: "Water flows into a tank at a rate r(t) gallons per minute. Use a midpoint Riemann sum with n=4 subintervals to approximate the total amount of water that flows in from t=0 to t=8 minutes."

Step 1: Find Δx = (8-0)/4 = 2 minutes.

Step 2: The midpoints are at t = 1, 3, 5, 7 minutes. These are x̄_i = 0 + (i − 1/2)*2 = 2i - 1.

Step 3: The sum in sigma notation is Σ_{i=1}^{4} r(2i - 1) * 2. If the rate table gives r(1)=3, r(3)=5, r(5)=7, r(7)=6, the approximation is (3+5+7+6)*2 = 42 gallons.

The units of the sum are (gallons/minute) * (minutes) = gallons, which matches the total amount. This is the standard AP Calculus AB FRQ rate table problem format. The scoring guidelines award one point for the correct setup in sigma notation and one point for the numerical value.

Trapezoidal Rule and Simpson's Rule Side by Side

The trapezoidal rule and Simpson's rule are common alternatives to the basic left, right, and midpoint sums. The trapezoidal rule averages the left and right sums for each subinterval, giving T_n = (Δx/2)[f(x_0) + 2f(x_1) + ... + 2f(x_{n-1}) + f(x_n)]. Simpson's rule uses a parabolic arc through three points, requiring an even number of subintervals, with S_n = (Δx/3)[f(x_0) + 4f(x_1) + 2f(x_2) + ... + 4f(x_{n-1}) + f(x_n)]. The basic Riemann sum formulas are covered here, while each rule has its own page.