The Riemann Sum Rule: Approximating the Area Under a Curve in Integral Calculus

Riemann Sum rule

The Riemann Sum Rule is a fundamental concept in integral calculus that helps us approximate the area under a curve

The Riemann Sum Rule is a fundamental concept in integral calculus that helps us approximate the area under a curve. It allows us to estimate the exact value of an integral by partitioning the interval into smaller subintervals and evaluating the function at specific points within each subinterval.

To understand the Riemann Sum Rule, let’s consider a function f(x) defined on a closed interval [a, b]. We want to estimate the definite integral of f(x) from a to b, denoted as ∫[a, b] f(x) dx.

The Riemann Sum Rule states that if we partition the interval [a, b] into n equally sized subintervals, denoted by Δx, and choose a point within each subinterval (usually the right endpoint or midpoint), we can approximate the integral using the summation:

∑[i=1 to n] f(x_i)Δx

Here, x_i represents the chosen point within each subinterval.

As the number of subintervals increases (approaching infinity), the Riemann sum approaches the exact value of the integral. This means that the approximation becomes more accurate as we use smaller subintervals and evaluate the function at more points.

There are different ways to evaluate Riemann sums depending on the type of partition and the choice of points within each subinterval. Some commonly used methods include the Left Riemann Sum (evaluating at the left endpoint), Right Riemann Sum (evaluating at the right endpoint), and Midpoint Riemann Sum (evaluating at the midpoint).

Here’s an example to illustrate how the Riemann Sum Rule works:

Let’s say we want to estimate the area under the curve y = x^2 on the interval [0, 2] using four subintervals.

First, we need to calculate the width of each subinterval, Δx. In this case, since we have four subintervals, Δx = (2 – 0)/4 = 0.5.

Next, we choose a point within each subinterval. Let’s use the right endpoints of each subinterval, which means the chosen points will be 0.5, 1.0, 1.5, and 2.0.

Now, we can write the Riemann sum:

∑[i=1 to 4] f(x_i)Δx = f(0.5)Δx + f(1.0)Δx + f(1.5)Δx + f(2.0)Δx

In this case, f(x) = x^2, so the Riemann sum becomes:

(0.5^2)(0.5) + (1.0^2)(0.5) + (1.5^2)(0.5) + (2.0^2)(0.5) = 0.125 + 0.5 + 1.125 + 2 = 3.75

So, the Riemann sum gives us an estimate of 3.75 for the area under the curve y = x^2 on the interval [0, 2] using four subintervals.

Remember that the more subintervals we use, the more accurate our approximation will be. As we increase the number of subintervals towards infinity, the Riemann sums will converge to the exact value of the integral.

Overall, the Riemann Sum Rule is a useful tool in calculus to approximate the area under a curve and evaluate definite integrals. It forms the foundation for more advanced integration techniques like the definite integral and Riemann Integral.

More Answers: