Sigma Notation with Left Endpoint- On Subinterval Approximation

What Sigma Notation Actually Is

Sigma notation is just a compact way to write long sums. Instead of writing x₁ + x₂ + x₃ + ... + x₁₀₀, you write Σ with the starting point, ending point, and what you're adding.

The symbol Σ (Greek letter sigma) means "add up everything that follows." It's not complicated. It's shorthand.

Why Left Endpoint Approximation Matters

When you can't find an exact integral, you approximate it using rectangles. The left endpoint rule builds rectangles using the left side of each subinterval as the rectangle's height.

This is the most intuitive approach. You look at each small chunk of the x-axis, grab the function value on its left edge, and draw a rectangle there.

The General Formula

For approximating ∫f(x)dx from a to b using n subintervals:

Δx = (b - a) / n

Approximation = Σ f(xᵢ) Δx, where i goes from 0 to n-1

The left endpoint of subinterval i is xᵢ = a + i·Δx

You evaluate f at each left endpoint, multiply by the width, and sum everything up.

Step-by-Step: How To Calculate Left Endpoint Approximation

Step 1: Find Δx

Divide your interval width by the number of rectangles.

Δx = (upper bound - lower bound) / number of subintervals

Step 2: Identify the left endpoints

Your x-values are: x₀, x₁, x₂, ... xₙ₋₁

Each one is: xᵢ = a + i·Δx

You stop at n-1 because the leftmost point of the last rectangle is the (n-1)th point.

Step 3: Evaluate the function at each left endpoint

Plug every xᵢ into f(x). Write down each result.

Step 4: Multiply by Δx and sum

Take each f(xᵢ), multiply by Δx, add them all together. That's your approximation.

Worked Example

Problem: Approximate ∫x² dx from 0 to 2 using n=4 subintervals with the left endpoint rule.

Step 1: Calculate Δx

Δx = (2 - 0) / 4 = 0.5

Step 2: Find left endpoints

x₀ = 0 + 0·0.5 = 0

x₁ = 0 + 1·0.5 = 0.5

x₂ = 0 + 2·0.5 = 1

x₃ = 0 + 3·0.5 = 1.5

Step 3: Evaluate f(x) = x² at each point

f(0) = 0² = 0

f(0.5) = 0.5² = 0.25

f(1) = 1² = 1

f(1.5) = 1.5² = 2.25

Step 4: Multiply and sum

Approximation = [0 + 0.25 + 1 + 2.25] × 0.5

Approximation = 3.5 × 0.5 = 1.75

The actual value is 8/3 ≈ 2.67, so our left endpoint approximation gives us an underestimate. More on that shortly.

Left vs. Right vs. Midpoint: The Comparison

Here is how left endpoint stacks up against other rectangle methods:

The left endpoint rule consistently underestimates increasing functions and overestimates decreasing functions. That's not a bug—it's information you can use.

When Left Endpoint Makes Sense

Left endpoint approximation is useful when:

• You want the simplest possible method to implement in code
• The function is strictly increasing and you need a conservative estimate
• You're doing quick preliminary calculations
• You want to pair with right endpoint to bound the true value

The left and right endpoint errors have opposite signs. If you average them, you get something close to the trapezoidal rule. If you average them and compare to the actual value, you know your error is bounded.

Common Mistakes

Using n points instead of n-1: With n subintervals, you only evaluate at n left endpoints, not n+1. The last left endpoint is xₙ₋₁, not xₙ.

Forgetting to multiply by Δx: The function values alone aren't the area. You must multiply each by the subinterval width before summing.

Confusing the formula for xᵢ: Left endpoints use xᵢ = a + i·Δx. Right endpoints use xᵢ = a + (i+1)·Δx. Different starting index.

Writing It in Sigma Notation

The full left endpoint approximation in sigma notation looks like this:

∫f(x)dx ≈ Σ f(a + i·Δx) · Δx where i = 0 to n-1

That's it. You can leave it in this form or expand it fully:

≈ [f(a) + f(a+Δx) + f(a+2Δx) + ... + f(a+(n-1)Δx)] · Δx