Problem 25 - Calculus Early Transcendentals: Pearson New International Edition

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.

Step-by-Step Solution

Answer

Left Riemann < Midpoint < Trapezoidal < Right Riemann.

Step 1: Understanding the Function

The function we are integrating is \( \sqrt{x^2 + 1} \) over the interval \( [0, 1] \). This function is increasing throughout the interval since as \( x \) increases, \( x^2 + 1 \) also increases, leading to an increase in \( \sqrt{x^2 + 1} \).

Step 2: Left Riemann Sum

The left Riemann sum uses the endpoints on the left of each subinterval to approximate the area under the curve. For a monotonically increasing function like \( \sqrt{x^2 + 1} \), it will underestimate the actual integral because it uses the lower-function-value point of the interval.

Step 3: Right Riemann Sum

Contrary to the left sum, the right Riemann sum uses the endpoints on the right of each subinterval. It overestimates the integral for increasing functions because it takes the higher-function-value point of the interval.

Step 4: Midpoint Riemann Sum

The midpoint Riemann sum takes the function values at the midpoints of each subinterval. For a smoothly increasing function, this usually offers a better approximation than both the left and right sums but depends on the curvature of the function.

Step 5: Trapezoidal Rule

The Trapezoidal Rule calculates the area by taking the average of the left and right Riemann sums. It usually provides a better approximation than both the left and right sums as it considers the progression of the function across the interval, and for many functions, it's better than the midpoint rule, especially for smoother curves.

Step 6: Ranking the Approximations

For this increasing and concave function, the approximations in order of smallest to largest are: Left Riemann sum < Midpoint sum < Trapezoidal Rule < Right Riemann sum. The left sum is smallest due to underestimation, the right is largest due to overestimation, and the trapezoidal is often more accurate than the midpoint.

Key Concepts

Riemann Sum

When learning about numerical integration, the Riemann Sum is a fundamental concept. It involves approximating the area under a curve. How does it work? It breaks the interval into smaller subintervals and then uses heights from either the left, right, or midpoint of these subintervals to construct rectangles.

• **Left Riemann Sum:** Uses the value at the left end, leading to an underestimate for increasing functions.
• **Right Riemann Sum:** Uses the value at the right end, causing an overestimate in increasing functions.
• **Midpoint Riemann Sum:** Utilizes the midpoint's value, often yielding a better balance between under- and overestimations.

The choice of whether to use the left, right, or midpoint can impact the accuracy based on the function's behavior over the interval.

Trapezoidal Rule

The Trapezoidal Rule is an exciting method for approximating definite integrals. It differs from the Riemann Sum because it takes into account the shape of the curve more accurately. Instead of using just rectangles, it uses trapezoids to roughly sketch the area between the curve and the x-axis.

• Each subinterval forms a trapezoid, with the function values at the endpoints as the top corners of the trapezoid.
• This rule averages out the heights from the left and right, which often provides a better approximation for smooth functions.

Due to this characteristic, the Trapezoidal Rule frequently offers a superior estimate compared to just employing the left or right Riemann sums. It accounts for changes in slopes and curvature more comprehensively.

Increasing Functions

Increasing functions, like the one in the original exercise, where the function \( \sqrt{x^2 + 1} \) is increasing over its interval, have distinct properties relevant in estimation methods.

• For increasing functions, the area under the curve steadily grows as x increases.
• The left Riemann Sum, typically using lower function values, tends to underestimate the actual area.
• The right Riemann Sum, often overestimates because it picks higher placement values.

Understanding these properties is crucial in determining the sequence of accuracy among approximation methods—an important aspect in numerical analysis.

Estimation Methods

Numerical integration involves using estimation methods to approximate the value of a definite integral. Various methods offer different levels of precision based on the function's nature and the desired accuracy.

Common Estimation Methods:

• **Riemann Sums** (Left, Right, Midpoint): Well-suited for different scenarios based on interpretation and curvature.
• **Trapezoidal Rule:** Provides good accuracy, especially for functions with linear or smooth curvatures over an interval.
• **Simpson's Rule** (not discussed here): Excellent for functions with higher curvature, more advanced.

Selecting the suitable method relies on understanding both the properties of the function you are working with and the precision requirement you aim to satisfy. This makes choosing an appropriate method crucial when solving real-world problems.