Problem 1a: Suppose We Divide The Interval 1 To 4 Into 100 P

Pblem 1a Suppose We Divide The Interval 1 4 Into 100 Equally Wid

Suppose we divide the interval [1, 4] into 100 equally wide subintervals and calculate a Riemann sum for f(x) = 1 + x² by randomly selecting a point cᵢ in each subinterval. The question asks for the maximum possible deviation of the Riemann sum from the exact area under the curve over this interval. Additionally, we explore the case when the interval is divided into 200 subintervals. The goal is to understand how the approximation improves as the number of subintervals increases, specifically in terms of the bounds of error in the Riemann sums.

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The task involves analyzing the error bounds of Riemann sums as an approximation of the definite integral of the function f(x) = 1 + x² over the interval [1, 4], when partitioned into subintervals of equal width. This analysis hinges on understanding both the properties of the function and the principles underpinning the Riemann sum errors.

First, the function f(x) = 1 + x² is continuous and differentiable over [1, 4], which allows us to apply the Mean Value Theorem and error estimation techniques associated with Riemann sums. The width Δx of each subinterval when dividing [1, 4] into n equal parts is given by:

Δx = (4 - 1) / n = 3 / n

For n = 100, Δx = 3/100 = 0.03; for n = 200, Δx = 3/200 = 0.015.

To find the maximum deviation of the Riemann sum from the actual area, the error bound associated with a Riemann sum (specifically, the lower and upper sums) can be estimated using the Mean Value Theorem for integrals and the properties of the function's derivatives. The key tool here is the error estimation for Riemann sums involving the second derivative of the function.

The second derivative of f(x) = 1 + x² is:

f''(x) = 2

This constant second derivative indicates the function is convex, and the error bounds for the integral approximation are proportional to the maximum of |f''(x)| over the interval and the square of the subinterval width:

|Error| ≤ (M / 2) * (Δx)²

where M is the maximum of |f''(x)| over [1, 4]. Since f''(x) = 2 everywhere, M = 2.

Thus, the error bound becomes:

|Error| ≤ (2 / 2) * (Δx)² = (Δx)²

For n = 100:

|Error| ≤ (0.03)² = 0.0009

For n = 200:

|Error| ≤ (0.015)² = 0.000225

But note that these bounds give a sense of the maximum possible deviation of the Riemann sum from the actual integral, assuming the most unfavorable case where the sampled points of the Riemann sum produce the largest error. Therefore, the Riemann sum with 100 subintervals will be within approximately 0.0009 of the true value, and with 200 subintervals, within approximately 0.000225.

This analysis demonstrates how increasing the number of subdivisions in the Riemann sum improves the approximation, reducing the potential error quadratically with respect to the subinterval width. As n increases, the bound on the error decreases significantly, confirming the convergence of Riemann sums to the exact integral value.

In conclusion, when dividing [1, 4] into 100 equally wide subintervals, the Riemann sum will be within about 0.0009 of the exact area under f(x). Doubling the number of subdivisions to 200 reduces this maximum deviation to about 0.000225, exemplifying the principle that finer partitions lead to more accurate integral approximations.

References

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