Left-endpoint approximation

Left-endpoint approximation is a Riemann-sum method in Calculus I that estimates area under a curve using the function value at each subinterval’s left endpoint as the rectangle height.

What is left-endpoint approximation?

Left-endpoint approximation is a way to estimate the area under a curve in Calculus I by building rectangles across an interval and using the left side of each subinterval to choose the height. Instead of trying to find exact area right away, you slice the interval into equal widths and add up the rectangle areas.

If the interval is divided into n parts, each part has width Δx. For each rectangle, you evaluate the function at the left endpoint of that subinterval, then multiply that value by Δx. The sum of those products is the left Riemann sum, often written as L_n = Σ f(x_i)Δx, with x_i taken from the left edge of each slice.

The word “approximation” matters here because the rectangles usually do not match the curve perfectly. If the function is increasing, left-endpoint rectangles sit below the curve for each slice, so the sum usually underestimates the true area. If the function is decreasing, those same left endpoints give heights that are too large, so the sum usually overestimates the area.

A quick example makes the pattern easier to see. Suppose you want the area under a positive, increasing function from x = 0 to x = 4 with 4 equal subintervals. Then Δx = 1, and the left endpoints are 0, 1, 2, and 3. You would compute f(0)(1) + f(1)(1) + f(2)(1) + f(3)(1), which gives an estimate based on the first point in each interval.

The main skill is setting up the rectangles correctly. A common mistake is using the right endpoint instead of the left endpoint, or forgetting that each rectangle needs the same width when the interval is split evenly. As you increase n, the rectangles get thinner, the estimate gets closer to the real area, and the sum starts looking more like the definite integral it is leading toward.

Why left-endpoint approximation matters in Calculus I

Left-endpoint approximation is one of the first places Calculus I connects graphing to integration. Before you get the exact definite integral, you need a way to estimate accumulated area from a function, and this method shows how sums of simple shapes can model curved regions.

It also builds your intuition for why the definite integral works. When the number of subintervals grows, each rectangle becomes narrower, so the gap between the curve and the approximation usually shrinks. That idea shows up again when you study Riemann sums, sigma notation, and the limit process behind integration.

This term also trains you to read a graph in a practical way. You are not just seeing a curve, you are deciding how to sample it, where to measure it from, and whether your estimate should be high or low based on whether the function rises or falls. That kind of reasoning shows up in problems where you estimate distance, total change, or accumulated quantity from data or a graph.

Left-endpoint approximation is especially useful because it gives you a clean first method to compare with other rectangle rules. Once you can spot what left endpoints do, midpoint and right-endpoint methods make more sense, and you can explain why one estimate is closer than another on a given graph.

How left-endpoint approximation connects across the course

Right-endpoint Approximation

Right-endpoint approximation uses the right side of each subinterval to set the rectangle height. It is the natural comparison point for left-endpoint approximation because both methods use the same interval split and the same rectangle widths, but they sample different points. On an increasing function, right-endpoint sums usually overshoot where left-endpoint sums undershoot.

Midpoint Approximation

Midpoint approximation chooses the function value halfway through each subinterval instead of at an endpoint. That difference often makes it more accurate than left-endpoint sums when the curve is smooth, because the sample point sits more centrally in each rectangle. It is a good next step once you understand how endpoint choice changes the estimate.

Δx

Δx is the width of each subinterval, and it tells you how wide every rectangle is in the approximation. Without Δx, you only have function values, not area estimates. In left-endpoint approximation, every term is a height times Δx, so getting the interval split correct is just as important as choosing the left endpoints.

sigma notation

Sigma notation is how you write the repeated addition in a compact way. Instead of listing every rectangle area one by one, you can write the whole left-endpoint sum as a single expression. That matters in Calculus I because Riemann sums are usually written in sigma form before you take limits and move toward the definite integral.

Is left-endpoint approximation on the Calculus I exam?

A problem set question will usually give you a function, an interval, and a number of subintervals, then ask for a left-endpoint estimate. Your job is to find Δx, list the left endpoints, evaluate the function at each one, and multiply by Δx before adding the results. If the graph is increasing or decreasing, you may also need to tell whether the estimate is an overestimate or underestimate. On quiz questions, the trick is often noticing that "left endpoint" means the sample point comes from the start of each subinterval, not the middle or the right side. You may also be asked to write the sum in sigma notation or compare it with a midpoint or right-endpoint approximation.

Left-endpoint approximation vs Right-endpoint Approximation

These two methods look almost identical because both use rectangles of equal width, but the sampled point changes. Left-endpoint approximation uses the first point in each subinterval, while right-endpoint approximation uses the last point. That difference changes whether the estimate tends to fall above or below the true area when the function is monotonic.

Key things to remember about left-endpoint approximation

• Left-endpoint approximation estimates area under a curve by using the function value at the left side of each subinterval.

• The rectangle width is Δx, so every rectangle area is f(left endpoint) times Δx.

• If the function is increasing, left-endpoint sums usually underestimate the true area.

• If the function is decreasing, left-endpoint sums usually overestimate the true area.

• The estimate gets closer to the actual area as you increase the number of subintervals.

Frequently asked questions about left-endpoint approximation

What is left-endpoint approximation in Calculus I?

It is a method for estimating area under a curve by cutting the interval into equal pieces and using the left endpoint of each piece to set the rectangle height. The total is a left Riemann sum. It gives you an approximation, not the exact area.

How do you find a left-endpoint approximation?

First find Δx by dividing the interval length by the number of subintervals. Then use the left endpoint of each subinterval to evaluate the function, multiply each value by Δx, and add the results. If the interval is [a, b] with n slices, the left endpoints start at a and move by Δx each time.

Is left-endpoint approximation an overestimate or underestimate?

It depends on the shape of the function. For an increasing function, left-endpoint rectangles are usually too short, so the sum is an underestimate. For a decreasing function, they are usually too tall, so the sum is an overestimate.

What is the difference between left-endpoint and midpoint approximation?

Left-endpoint approximation samples the function at the start of each subinterval, while midpoint approximation samples it halfway across each one. Midpoint often gives a better estimate because it balances the rectangle more evenly around the curve. The setup is similar, but the chosen x-values change the result.