Left-endpoint approximation

Definition

Left-endpoint approximation is a method used to estimate the area under a curve on a given interval by summing the areas of rectangles whose heights are determined by the function's value at the left endpoints of subintervals.

5 Must Know Facts For Your Next Test

1. The left-endpoint approximation involves dividing the interval into $n$ subintervals of equal width, $\Delta x$.
2. The height of each rectangle in this method is determined by evaluating the function at the left endpoint of each subinterval.
3. The formula for left-endpoint approximation is $L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x$, where $x_i$ represents the left endpoint of each subinterval.
4. This method tends to underestimate or overestimate the true area depending on whether the function is increasing or decreasing on the interval.
5. The accuracy of the approximation improves as the number of subintervals increases.

Review Questions

• How do you determine the height of each rectangle in a left-endpoint approximation?
• What is the formula for calculating left-endpoint approximation?
• How does increasing the number of subintervals affect the accuracy of a left-endpoint approximation?

Related terms

Right-endpoint Approximation: A method for estimating area under a curve where rectangles' heights are determined by function values at right endpoints of subintervals.

Midpoint Approximation: A technique for approximating area under a curve using rectangles whose heights are determined by function values at midpoints of subintervals.

$\Delta x$: $\Delta x$ represents the width of each subinterval when dividing an interval into equal parts for numerical integration methods.