Question

Without doing any calculations, rank from smallest to largest the approximations of \(\int_{0}^{1} \sqrt{x^{2}+1} d x\) for the following methods: left Riemann sum, right Riemann sum, midpoint Riemann sum, Trapezoidal Rule.

Short answer

Left < Midpoint < Trapezoidal < Right

Step-by-step solution

Understand the Function

The integral \(\int_{0}^{1} \sqrt{x^{2}+1} \ dx\) represents the area under the curve of the function \(f(x) = \sqrt{x^{2}+1}\) from \(x = 0\) to \(x = 1\). This function is increasing on \([0, 1]\), since the derivative \(f'(x) = \frac{x}{\sqrt{x^{2}+1}}\) is always non-negative.

Analyze the Left Riemann Sum

The Left Riemann Sum estimates the area under the curve using rectangles whose height is the value of the function at the left endpoint of each subinterval. Given that \(f(x)\) is increasing, the Left Riemann Sum will underestimate the integral because the left endpoint of each subinterval underestimates \(f(x)\) throughout the interval.

Analyze the Right Riemann Sum

The Right Riemann Sum uses rectangles whose height is the value of the function at the right endpoint of each subinterval. Since \(f(x)\) is increasing, the Right Riemann Sum will overestimate the integral because the right endpoint of each subinterval overestimates \(f(x)\) throughout the interval.

Analyze the Midpoint Riemann Sum

The Midpoint Riemann Sum uses rectangles whose height is the value of the function at the midpoint of each subinterval. This often gives a more accurate estimate than simple left or right sums, as it balances overestimations and underestimations, being closer to the actual value of the integral, especially for continuous functions.

Analyze the Trapezoidal Rule

The Trapezoidal Rule approximates the area under the curve using trapezoids instead of rectangles, which averages the left and right endpoints. It provides a better approximation than the Left or Right Riemann Sum because it compensates for both underestimation and overestimation.

Rank the Approximations

As the function is increasing on \([0,1]\), the Left Riemann Sum will provide the smallest approximation, followed by the Midpoint, the Trapezoidal Rule, and finally, the Right Riemann Sum will provide the largest approximation.

Key concepts

Integral Approximations

When we talk about integral approximations, we are trying to estimate the area under a curve where finding an exact integral might be complicated or impossible to compute directly. We use methods like Riemann sums and the Trapezoidal Rule for these approximations.
For an increasing function on a specific interval, the Riemann sums give us a way to gauge the area using simpler geometric shapes like rectangles or trapezoids. This is useful in cases where the function doesn't have an antiderivative, or when it's cumbersome to compute it.
We use three primary types of Riemann sums: Left, Right, and Midpoint. Each has its unique way of estimating the area, depending on where the height of the rectangle is taken in the subinterval. The choice of method impacts the accuracy of the approximation. Understanding how each method affects the estimation helps in choosing the right one for various integral calculations.

Trapezoidal Rule

The Trapezoidal Rule is a more refined method of approximating integrals compared to elementary Riemann sums. Instead of using rectangles, this method uses trapezoids to approximate the area under a curve. Each subinterval is represented by a trapezoid rather than a rectangle.
This trapezoid is formed using the function's value at the endpoints of each subinterval, which are connected to create a slanted line.
By averaging the results of the left and right estimates, the Trapezoidal Rule tends to provide a better approximation of the integral. This is especially true for smooth and continuous functions, where linear interpolation between endpoints captures the curve more accurately. It tends to be more accurate than the Left or Right Riemann sums but can be less precise than the Midpoint method for certain functions.

Increasing Functions

An understanding of increasing functions is crucial when applying numerical integration methods. A function is said to be increasing on an interval if, for any two numbers in that interval, a larger input number results in a larger output number.
For the function given in the exercise, \(f(x) = \sqrt{x^{2}+1}\), the derivative \(f'(x) = \frac{x}{\sqrt{x^{2}+1}}\) is greater than or equal to zero on \([0,1]\), confirming it is an increasing function on this interval.
This behavior affects how the integral approximations work. With increasing functions, the Left Riemann sum will typically underestimate the integral, and the Right Riemann sum will overestimate it. The predictability of these traits for increasing functions allows us to rank these approximation methods even without explicit calculations.

Numerical Integration Methods

Numerical integration encompasses a variety of methods used to estimate the value of a definite integral. When direct integration isn't feasible, these methods serve as powerful tools to find approximate solutions.
Some of the most commonly used methods include Riemann sums, the Trapezoidal Rule, and Simpson's Rule, each with its own approach and level of precision.
When dealing with increasing functions, methods like the Midpoint sum often provide better accuracy than simple Left or Right Riemann sums because they balance errors more effectively. These techniques are not just theoretically important, but also practically beneficial when solving real-world problems where exact solutions are elusive. Understanding the strengths and limitations of each method helps in determining the most suitable one for particular functions and intervals.