Midpoint Riemann Sum with Summation on Calculator- Step-by-Step

What the Heck Is a Midpoint Riemann Sum?

You have an integral to evaluate. You don't want to do it by hand. Your calculator can handle it, but you need to know the right sequence of buttons.

A midpoint Riemann sum divides your interval into rectangles. Instead of using the left or right endpoint of each subinterval, you use the midpoint. It's more accurate than left or right endpoints, but still an approximation unless you're summing infinitely many rectangles.

The formula:

Midpoint Sum = Δx × [f(x₁*) + f(x₂*) + f(x₃*) + ... + f(xₙ*)]

Where:

• Δx = (b - a) / n (width of each rectangle)
• n = number of rectangles
• xᵢ* = midpoint of each subinterval
• a = lower bound
• b = upper bound

Step-by-Step: Midpoint Riemann Sum on Your Calculator

Let's say you need to evaluate the midpoint sum for f(x) = x² from x = 0 to x = 4 with n = 4 rectangles.

Step 1: Find Δx

Δx = (4 - 0) / 4 = 1

Each rectangle has width 1.

Step 2: Find the Midpoints

Your subintervals are [0,1], [1,2], [2,3], [3,4].

The midpoints are:

• Subinterval 1: (0 + 1) / 2 = 0.5
• Subinterval 2: (1 + 2) / 2 = 1.5
• Subinterval 3: (2 + 3) / 2 = 2.5
• Subinterval 4: (3 + 4) / 2 = 3.5

Step 3: Evaluate f(x) at Each Midpoint

f(0.5) = 0.25

f(1.5) = 2.25

f(2.5) = 6.25

f(3.5) = 12.25

Step 4: Enter the Summation

On a TI-84 Plus:

Press 2nd → STAT (above the math button) → scroll to seq( → ENTER

The syntax is:

seq(expression, variable, start, end, step)

For our example, each midpoint follows the pattern: 0.5 + i, where i starts at 0 and increments by 1.

Enter this:

seq(X², X, 0.5, 3.5, 1)

Press ENTER. You'll get: {0.25, 2.25, 6.25, 12.25}

Step 5: Sum the Results

Store the sequence in a list or use the sum function.

Press 2nd → STAT → Math → sum(

Then enter your sequence:

sum(seq(X², X, 0.5, 3.5, 1))

Result: 21

Step 6: Multiply by Δx

21 × 1 = 21

That's your midpoint Riemann sum approximation.

Using the Summation Function Directly

Most TI-84s have a summation template. Press 2nd → STAT → scroll to sum(. But you need the Σ( function for symbolic summation.

Actually, the built-in summation function is buried deeper:

Press 2nd → 0 (for CATALOG) → scroll down to fnInt( — wait, that's for integrals. For summation, use the sequence method above. It works every time.

Quick Reference: TI-84 Button Sequence

Common Mistakes That Will Kill Your Answer

• Using endpoints instead of midpoints. This gives you left or right Riemann sums, not midpoint. Check your subinterval calculation.
• Forgetting to multiply by Δx. The sum of f(xᵢ*) values is not your final answer. You must multiply by the rectangle width.
• Wrong n value. If you use n=4 rectangles, you need 4 midpoints, not 5. Common off-by-one error.
• Calculator in degree mode. If your function involves trig, make sure your calculator is in radian mode. Press MODE → scroll to RADIAN → ENTER.
• Syntax errors in seq(). The format is seq(expression, variable, lower, upper, step). Don't mix up the order.

How to Check If Your Answer Is Reasonable

The midpoint sum should be closer to the actual integral than left or right sums (unless your function is linear, in which case all methods give the exact answer).

For f(x) = x² from 0 to 4:

• Actual integral: ∫₀⁴ x² dx = 64/3 ≈ 21.33
• Midpoint sum with n=4: 21 ✓ (pretty close)
• Left sum: 14 (underestimates badly)
• Right sum: 30 (overestimates badly)

If your midpoint sum is way off from the actual integral, something went wrong in your setup.

Using More Rectangles for Better Accuracy

Increase n for a better approximation. Let's try n = 8 for the same problem:

Δx = (4 - 0) / 8 = 0.5

Midpoints: 0.25, 0.75, 1.25, 1.75, 2.25, 2.75, 3.25, 3.75

Enter: sum(seq(X², X, 0.25, 3.75, 0.5))

Result: 42

42 × 0.5 = 21

Wait — same answer? Let's verify. Actually, for x², the midpoint rule with symmetric subintervals sometimes gives the exact result for certain values. In general, more rectangles = better approximation.

Try n=10:

sum(seq(X², X, 0.2, 3.8, 0.4))

Multiply by 0.4 = 21.28

Getting closer to 21.33.

The Bottom Line

Midpoint Riemann sums on a calculator come down to three things:

1. Calculate your midpoints correctly: a + Δx/2, then add Δx repeatedly
2. Use seq() to evaluate your function at each midpoint
3. Use sum() to add them up, then multiply by Δx

That's it. No magic. Just arithmetic and knowing where the buttons are.