Understanding Right-Endpoint Approximation: A Method for Approximating the Area Under a Curve

Right-Endpoint Approximation

Right-Endpoint Approximation, also known as Right Riemann Sum, is a method used to approximate the area under a curve by dividing it into several smaller rectangles and summing their areas

Right-Endpoint Approximation, also known as Right Riemann Sum, is a method used to approximate the area under a curve by dividing it into several smaller rectangles and summing their areas. This technique is often used in numerical integration when the exact value of the integral is difficult to calculate.

To understand right-endpoint approximation, let’s consider a function f(x) that is defined on the interval [a, b]. We want to approximate the area under the curve of this function on that interval.

First, we divide the interval [a, b] into n equal subintervals, each of width Δx. The width Δx is calculated as Δx = (b – a) / n.

Now, for each subinterval, we choose the right endpoint as our x-value for calculating the height of the rectangle. This means that for the first subinterval, we choose x = a + Δx, for the second subinterval, we choose x = a + 2Δx, and so on. This way, we are evaluating the function at the right endpoint of each subinterval.

Next, we calculate the height of each rectangle by evaluating the function f at these x-values. Let’s say the height of the rectangle for the i-th subinterval is f(xi), where xi is the right endpoint of that subinterval.

Finally, we calculate the area of each rectangle by multiplying its height, f(xi), with the width, Δx. The area of the rectangle for the i-th subinterval is then given by f(xi) * Δx.

To approximate the total area under the curve, we sum up the areas of all the rectangles. This can be done by adding up the areas of each individual rectangle:

Approximated area = f(x1) * Δx + f(x2) * Δx + … + f(xn) * Δx

In mathematical notation, we can write the right-endpoint approximation as:

Approximated area ≈ Σ f(xi) * Δx for i = 1 to n

As the number of subintervals increases, i.e., as n approaches infinity, the right-endpoint approximation will become more accurate and will approach the actual area under the curve.

It’s important to note that right-endpoint approximation is just one of the many methods used to approximate the area under a curve. Other methods include left-endpoint approximation, midpoint approximation, and trapezoidal approximation, each with its own calculations and level of accuracy.

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