Left Reimann Sum
The left Riemann sum is a method used to approximate the value of a definite integral using rectangles, specifically by evaluating the function at the left endpoint of each subinterval and multiplying it by the width of the subinterval
The left Riemann sum is a method used to approximate the value of a definite integral using rectangles, specifically by evaluating the function at the left endpoint of each subinterval and multiplying it by the width of the subinterval. This approximation becomes better as the number of subintervals increases.
To understand the left Riemann sum, let’s consider a function f(x) that is continuous on the interval [a, b], and we want to calculate the definite integral of f(x) from a to b. We divide the interval [a, b] into n subintervals of equal width Δx = (b – a)/n.
Now, we need to find the left endpoints of each subinterval. The left endpoint of the k-th subinterval is given by x_k = a + (k – 1)Δx, where k ranges from 1 to n.
To calculate the left Riemann sum, we evaluate the function f(x) at each left endpoint x_k, and then sum up the product of f(x_k) and Δx for all subintervals. Mathematically, the left Riemann sum can be expressed as:
L_n = Δx * [f(x_1) + f(x_2) + … + f(x_n-1) + f(x_n)]
Where L_n represents the left Riemann sum with n subintervals.
As mentioned earlier, as the number of subintervals increases, the left Riemann sum becomes a better approximation of the definite integral. In the limit as n approaches infinity, the left Riemann sum converges to the actual value of the definite integral.
It’s important to note that the left Riemann sum can be an overestimate or underestimate of the actual definite integral depending on the function and the behavior of the function between the left endpoints of the subintervals.
So, in summary, the left Riemann sum is a method for approximating the value of a definite integral by evaluating the function at the left endpoint of each subinterval and summing up the products with the width of the subinterval.
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