Calculate Exact Limit of The Rimann Sum As N

The exact limit of a Riemann sum as n approaches infinity is a fundamental concept in calculus that connects the idea of summation to integration. This limit, known as the definite integral, represents the area under a curve between two points. Understanding how to calculate this limit is essential for solving problems in physics, engineering, and economics.

What is a Riemann Sum?

A Riemann sum is a method for approximating the area under a curve by dividing the area into a series of rectangles. The sum of the areas of these rectangles provides an approximation of the exact area under the curve.

The general formula for a Riemann sum is:

R_n = Σ f(x_i*) Δx

where:

n is the number of rectangles
f(x_i*) is the value of the function at a sample point in the i-th subinterval
Δx is the width of each subinterval

The exact limit of this sum as n approaches infinity is the definite integral of the function over the interval [a, b].

Calculating the Limit of a Riemann Sum

To find the exact limit of a Riemann sum, we need to evaluate the definite integral of the function. The definite integral represents the exact area under the curve between the points a and b.

The formula for the definite integral is:

∫_a^b f(x) dx = lim_n→∞ Σ f(x_i*) Δx

This limit is the exact value of the area under the curve, and it's what we're trying to calculate.

Steps to Calculate the Limit

Identify the function f(x) and the interval [a, b].
Choose a sample point x* in each subinterval.
Calculate the width of each subinterval Δx = (b - a)/n.
Compute the Riemann sum R_n = Σ f(x_i*) Δx.
Take the limit as n approaches infinity to find the exact value.

Example Calculation

Let's calculate the exact limit of the Riemann sum for the function f(x) = x² on the interval [0, 1].

We'll use the right endpoint method for this example, where x* = x_i+1.

Step-by-Step Calculation

Divide the interval [0, 1] into n equal subintervals, each of width Δx = 1/n.
For each subinterval, take the right endpoint x_i = i/n.
Calculate the Riemann sum: R_n = Σ (i/n)² (1/n).
Simplify the sum: R_n = (1/n³) Σ i².
Use the formula for the sum of squares: Σ i² = n(n+1)(2n+1)/6.
Substitute and take the limit as n → ∞: lim_n→∞ (1/n³) [n(n+1)(2n+1)/6] = 1/3.

The exact limit of the Riemann sum is 1/3, which matches the definite integral ∫₀¹ x² dx.

Frequently Asked Questions

What is the difference between a Riemann sum and a definite integral?

A Riemann sum is an approximation of the area under a curve using rectangles. A definite integral is the exact limit of these Riemann sums as the number of rectangles approaches infinity.

How do I choose the sample point x* in each subinterval?

You can choose the left endpoint, right endpoint, or any point within the subinterval. Different choices can lead to different Riemann sums, but they all approach the same limit as n → ∞.

What happens if the function is not continuous?

If the function has a finite number of discontinuities, the Riemann sum still approaches the definite integral. However, if the function has infinite discontinuities, the Riemann sum may not converge.