Left-endpoint approximation

Definition

Left-endpoint approximation is a method for estimating the area under a curve by summing the areas of rectangles whose heights are determined by the function value at the left endpoints of subintervals. It provides an estimate that can be used to approximate definite integrals.

5 Must Know Facts For Your Next Test

1. The left-endpoint approximation formula is $L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x$, where $\Delta x$ is the width of each subinterval.
2. As the number of subintervals increases, the approximation generally becomes more accurate.
3. Left-endpoint approximations can underestimate or overestimate the true area depending on whether the function is increasing or decreasing.
4. To apply this method, divide the interval of integration into equal-width subintervals and use the function value at the start (left end) of each subinterval.
5. This method is part of a family of Riemann sums, which also include right-endpoint and midpoint approximations.

Review Questions

• What is the formula for calculating a left-endpoint approximation?
• How does increasing the number of subintervals affect the accuracy of a left-endpoint approximation?
• In what scenarios might a left-endpoint approximation underestimate or overestimate the area under a curve?

Related terms

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

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

Riemann Sum: A sum that approximates an integral by partitioning an interval into smaller segments and summing up areas calculated using sample points within those segments.