Understanding the Right Riemann Sum: A Method for Approximating the Area Under a Curve

right riemann sum

The right Riemann sum is a method in calculus used to approximate the area under a curve by dividing it into several rectangles and summing their areas

The right Riemann sum is a method in calculus used to approximate the area under a curve by dividing it into several rectangles and summing their areas. This approximation is based on the right endpoints of each interval.

To understand the right Riemann sum, let’s consider a function f(x) that is defined and non-negative on a closed interval [a, b]. We want to approximate the definite integral ∫[a, b] f(x) dx using rectangles.

First, we divide the interval [a, b] into n subintervals of equal width Δx = (b – a) / n. The right endpoints of these subintervals are given by x_i = a + iΔx, where i ranges from 1 to n.

Next, we construct rectangles, where the height of each rectangle is determined by the value of the function f(x) at the right endpoint x_i, and the width is given by Δx.

The right Riemann sum is then given by the sum of the areas of these rectangles:

R_n = Δx * (f(x_1) + f(x_2) + … + f(x_n-1) + f(x_n))

The area of each rectangle is the product of its height (f(x_i)) and width (Δx). Finally, by taking the limit as n approaches infinity, we can obtain a more accurate approximation of the area under the curve, represented by the definite integral ∫[a, b] f(x) dx.

Riemann sums, including the right Riemann sum, provide a useful tool in understanding and approximating the area under curves, even when analytical methods are not applicable or difficult to compute. It is important to note that the choice of endpoints (left, right, or even midpoints) affects the accuracy of the approximation, and using more rectangles generally leads to a better approximation.

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