Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

Many computer algebra systems permit the evaluation of Riemann sums for left end point, right end point, or midpoint evaluations of the function. Using such a system, evaluate the 10 -subinterval Riemann sums using left end point, right end point, and midpoint evaluations. $$ \int_{0}^{1} \tan x d x $$

Step-by-Step Solution

Answer

Use computer algebra systems for the calculations: L_10, R_10, and M_10 approximations.

Step 1: Understand the Setup

We have the integral \( \int_{0}^{1} \tan x \, dx \), and we need to approximate it using Riemann sums with 10 subintervals. The widths of these subintervals are calculated as \( \Delta x = \frac{b-a}{n} = \frac{1-0}{10} = 0.1 \).

Step 2: Compute Left Endpoint Approximation

For the left endpoint approximation, the sample points are \( x_0 = 0, x_1 = 0.1, \ldots, x_9 = 0.9 \). Calculate \( f(x_i) = \tan(x_i) \) for each \( i \) from 0 to 9. Sum them up and multiply by \( \Delta x \): \[ L_{10} = \sum_{i=0}^{9} \tan(x_i) \cdot 0.1 \].Using a computer algebra system:\( L_{10} = 0 + \tan(0.1) \cdot 0.1 + \tan(0.2) \cdot 0.1 + \ldots + \tan(0.9) \cdot 0.1 \).

Step 3: Compute Right Endpoint Approximation

For the right endpoint approximation, the sample points are \( x_1 = 0.1, x_2 = 0.2, \ldots, x_{10} = 1.0 \). Calculate \( f(x_i) = \tan(x_i) \) for each \( i \) from 1 to 10. Sum them up and multiply by \( \Delta x \):\[ R_{10} = \sum_{i=1}^{10} \tan(x_i) \cdot 0.1 \].Using a computer algebra system:\( R_{10} = \tan(0.1) \cdot 0.1 + \tan(0.2) \cdot 0.1 + \ldots + \tan(1.0) \cdot 0.1 \).

Step 4: Compute Midpoint Approximation

For the midpoint approximation, calculate the midpoints of each subinterval: \( x_{0.5} = 0.05, x_{1.5} = 0.15, \ldots, x_{9.5} = 0.95 \). Evaluate \( f(x_{i+0.5}) = \tan(x_{i+0.5}) \) for each \( i \) from 0 to 9. Sum them up and multiply by \( \Delta x \):\[ M_{10} = \sum_{i=0}^{9} \tan(x_{i+0.5}) \cdot 0.1 \].Using a computer algebra system:\( M_{10} = \tan(0.05) \cdot 0.1 + \tan(0.15) \cdot 0.1 + \ldots + \tan(0.95) \cdot 0.1 \).

Step 5: Calculate using Computer Algebra System

Using a computer algebra system like Maple, Mathematica, or an online calculator, plug in each set of sample points and calculate the sums for left, right, and midpoint approximations separately.

Key Concepts

Integral Approximation

Integral approximation is a method used to estimate the value of definite integrals, especially when calculating an exact integral is challenging. By using Riemann sums, we break the interval over which the integral is calculated into smaller subintervals. Each subinterval represents a segment over which we approximate the area under a curve defined by a function. The sum of these small areas gives us an estimate of the total area, which is the approximation of the integral. This method is highly useful in situations where finding a precise antiderivative is complex or impossible.

Subinterval Method

The subinterval method is a key part of the Riemann sum approach. Instead of trying to figure out the entire area under a curve at once, we divide the interval of interest into smaller pieces or subintervals. Each of these subintervals has a width calculated as:

• \( \Delta x = \frac{b-a}{n} \)

where \( b \) and \( a \) are the boundaries of the interval, and \( n \) is the number of subintervals we decide to use. In our example, we wanted to approximate the integral from 0 to 1, so we used 10 subintervals, each with a width of 0.1. This method is beneficial as it allows us to make our approximation more accurate by simply increasing the number of subintervals.

Left Endpoint Approximation

The left endpoint approximation is a specific type of integral approximation where we take the height of each rectangle from the left endpoint of each subinterval. It works by evaluating the function at the leftmost endpoint of each subinterval and using that function value as the rectangle's height.

• Sample Points: \( x_0, x_1, \ldots, x_{n-1} \)
• Formula: \[ L_{n} = \sum_{i=0}^{n-1} f(x_i) \cdot \Delta x \]

This approach can be advantageous when the function is decreasing over the interval, making the left endpoint provide a closer approximation.

Right Endpoint Approximation

The right endpoint approximation is similar to the left endpoint approximation, but here we use the rightmost point to calculate the height of each rectangle. This involves evaluating the function at the rightmost endpoint of each subinterval. This technique tends to provide a better approximation in cases where a function is increasing over the interval.

• Sample Points: \( x_1, x_2, \ldots, x_n \)
• Formula: \[ R_{n} = \sum_{i=1}^{n} f(x_i) \cdot \Delta x \]

By using this approach, one might get an overestimate or underestimate depending on whether the function is increasing or decreasing within the given interval.

Midpoint Approximation

The midpoint approximation stands out by calculating the height of each rectangular section at a point exactly halfway across each subinterval, known as the midpoint. This can provide a more balanced estimate when compared to left or right approximations since it accounts for both sides of the subinterval more equally.

• Midpoints: \( x_{0.5}, x_{1.5}, \ldots, x_{n-0.5} \)
• Formula: \[ M_{n} = \sum_{i=0}^{n-1} f(x_{i+0.5}) \cdot \Delta x \]

Using the midpoint approximation often results in a more accurate estimation of the integral, especially for well-behaved functions over small intervals.