Problem 4 - Calculus Early Transcendentals

Question

If the Midpoint Rule is used on the interval [-1,11] with \(n=3\) sub-intervals, at what \(x\) -coordinates is the integrand evaluated?

Step-by-Step Solution

Answer

Answer: The integrand is evaluated at the x-coordinates 1, 5, and 9.

Step 1: Determine the width of each sub-interval

To calculate the width of each sub-interval, we need to divide the total width of the interval by the number of sub-intervals (n). The width of the whole interval is: width = b - a = 11 - (-1) = 12 Now, we can calculate the width of each sub-interval: width of each sub-interval = width / n = 12 / 3 = 4

Step 2: Find the midpoints of each sub-interval

We are given 3 sub-intervals, and we know that each one has a width of 4 units. We can find the midpoints of each sub-interval by starting from the left endpoint of the interval (-1) and adding half of the width (2) to it. This will give us the midpoint of the first sub-interval. Midpoint of the first sub-interval: -1 + 2 = 1 To find the midpoint of the second sub-interval, we can simply add the width (4) to the midpoint of the previous sub-interval. Midpoint of the second sub-interval: 1 + 4 = 5 To find the midpoint of the third sub-interval, again, we can add the width (4) to the midpoint of the previous sub-interval. Midpoint of the third sub-interval: 5 + 4 = 9 The integrand is evaluated at the following x-coordinates: 1, 5, and 9.

Key Concepts

Numerical Integration

Numerical integration is a cornerstone of mathematical analysis, providing techniques to approximate the definite integral of a function when an analytical solution is difficult or impossible to obtain. One prominent method of numerical integration is the Midpoint Rule, an approach that lies under the umbrella of the Riemann sum. Instead of finding the exact area under a curve, we calculate an approximate value by summing the areas of rectangles, each representing a slice of the space below the curve. These rectangles are centered at the midpoint of sub-intervals along the x-axis, hence the name Midpoint Rule.

The essence of this method can be likened to estimating the total area of a uniquely shaped garden by placing equally sized, square mats throughout, and then considering the total area those mats cover. It's an efficient way to get a good approximation with relatively simple calculations, especially when the function in question behaves nicely within the interval.

Sub-Intervals in Integration

When using numerical methods like the Midpoint Rule, the continuous interval over which we are integrating—the 'garden'—is divided into smaller segments called sub-intervals. This division is a critical step as the number of sub-intervals, symbolized by 'n', and their width directly affect the accuracy of the integration.

Consider the task of planting new flowers in our garden: the flower beds (our sub-intervals) should be evenly spaced and of equal size to create a symmetrical layout. By having uniform sub-intervals, we ensure that each part of the function (or garden) is equally represented, which in turn helps achieve a more accurate approximation. As a rule of thumb, more sub-intervals generally lead to higher accuracy as we are considering more points along the function, but it also demands more computational effort.

Integrand Evaluation

The evaluation of the integrand—the function being integrated—at specific points within each sub-interval is the final step of the Midpoint Rule. It's akin to measuring the height of a plant at the center of each flower bed to estimate how much space the flowers will occupy.

In the given exercise, the function would be evaluated at the coordinates 1, 5, and 9. These points, found at the heart of each sub-interval, are the bastions of our approximation. They serve as the heights of our rectangles that, when multiplied by the width of the sub-intervals, provide us with an estimate of the integral. This process mirrors the way we aggregate more detailed information—like individual plant heights—to approximate the overall visual impact of the entire garden.