A left Riemann sum is a method for approximating the area under a curve by dividing the interval into subintervals and using the left endpoints of each subinterval to calculate the sum of the areas of rectangles. This technique connects to the fundamental concept of definite integrals, as it provides a way to estimate the total area under a function over a given interval by adding up these rectangle areas.
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In a left Riemann sum, each rectangle's height is determined by the function value at the left endpoint of each subinterval.
The approximation can be improved by increasing the number of subintervals, which makes each rectangle narrower and gives a better estimate of the area.
Left Riemann sums tend to underestimate the area under a curve if the function is increasing and overestimate if it is decreasing.
The formula for calculating a left Riemann sum can be expressed as $$L_n = ext{width} imes (f(x_0) + f(x_1) + ... + f(x_{n-1}))$$ where $$n$$ is the number of subintervals.
When computing a left Riemann sum, it's crucial to ensure that all subintervals are of equal width for consistency in estimation.
Review Questions
How does a left Riemann sum compare to a right Riemann sum in terms of estimating the area under a curve?
A left Riemann sum uses the function values at the left endpoints of subintervals, while a right Riemann sum utilizes the values at the right endpoints. When dealing with increasing functions, left Riemann sums generally underestimate the area, whereas right Riemann sums overestimate it. Conversely, for decreasing functions, left sums overestimate while right sums underestimate. This difference highlights how choosing endpoints affects approximation accuracy.
What is the relationship between left Riemann sums and definite integrals when considering limits and approximations?
Left Riemann sums serve as an approximation for definite integrals, which represent the exact area under a curve. As the number of subintervals increases and their widths decrease, the left Riemann sum approaches the value of the definite integral. This connection illustrates how these sums can be used to estimate areas before finding precise values through integration techniques. Understanding this relationship helps reinforce fundamental calculus concepts related to area calculation.
Evaluate how changing the number of subintervals affects the accuracy of a left Riemann sum and its convergence to a definite integral.
Increasing the number of subintervals in a left Riemann sum reduces the width of each rectangle, which improves the approximation's accuracy. As more subintervals are used, any errors caused by using only one endpoint diminish, leading to better convergence towards the actual value represented by a definite integral. This demonstrates not only how numerical methods can approximate areas but also emphasizes how limits play a vital role in calculus, bridging summation techniques with integration.
Related terms
Right Riemann Sum: A right Riemann sum approximates the area under a curve using the right endpoints of subintervals instead of the left endpoints.
A definite integral calculates the exact area under a curve between two specified points, serving as the limit of Riemann sums as the number of subintervals approaches infinity.
Subinterval: A subinterval is a smaller segment of the larger interval that is used in Riemann sums to approximate area under a curve.