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Chapter 7: Problem 13

Find the indicated Midpoint Rule approximations to the following integrals. \(\int_{0}^{1} \sin \pi x d x\) using \(n=6\) subintervals

Short Answer

Expert verified
Question: Using the Midpoint Rule with 6 subintervals, approximate the value of the integral \(\int_{0}^{1} \sin(\pi x) d x\). Answer: \(\int_{0}^{1} \sin(\pi x) d x \approx \frac{1}{6}\left[\sin\left(\pi \cdot \frac{1}{12}\right) + \sin\left(\pi \cdot \frac{1}{4}\right) + \sin\left(\pi \cdot \frac{5}{12}\right) + \sin\left(\pi \cdot \frac{7}{12}\right) + \sin\left(\pi \cdot \frac{3}{4}\right) + \sin\left(\pi \cdot \frac{11}{12}\right)\right]\)

Step by step solution

01

Identify the integral and the number of subintervals

Given integral: \(\int_{0}^{1} \sin(\pi x) d x\) Number of subintervals: \(n = 6\)
02

Calculate the width of each subinterval

The width of each subinterval is given by: \(\Delta x = \frac{b-a}{n} = \frac{1-0}{6} = \frac{1}{6}\)
03

Find the midpoints of each subinterval

Midpoints for each subinterval are given by: \(x_i = a + i \cdot \Delta x + \frac{1}{2} \Delta x\) for \(i=0, 1, \ldots, n-1\) \(x_0 = 0 + 0 \cdot \frac{1}{6} + \frac{1}{12} = \frac{1}{12}\) \(x_1 = 0 + 1 \cdot \frac{1}{6} + \frac{1}{12} = \frac{3}{12} = \frac{1}{4}\) \(x_2 = 0 + 2 \cdot \frac{1}{6} + \frac{1}{12} = \frac{5}{12}\) \(x_3 = 0 + 3 \cdot \frac{1}{6} + \frac{1}{12} = \frac{7}{12}\) \(x_4 = 0 + 4 \cdot \frac{1}{6} + \frac{1}{12} = \frac{9}{12} = \frac{3}{4}\) \(x_5 = 0 + 5 \cdot \frac{1}{6} + \frac{1}{12} = \frac{11}{12}\)
04

Evaluate \(\sin(\pi x)\) at each midpoint

We now evaluate the function at each midpoint: \(f(x_0) = \sin\left(\pi \cdot \frac{1}{12}\right)\) \(f(x_1) = \sin\left(\pi \cdot \frac{1}{4}\right)\) \(f(x_2) = \sin\left(\pi \cdot \frac{5}{12}\right)\) \(f(x_3) = \sin\left(\pi \cdot \frac{7}{12}\right)\) \(f(x_4) = \sin\left(\pi \cdot \frac{3}{4}\right)\) \(f(x_5) = \sin\left(\pi \cdot \frac{11}{12}\right)\)
05

Apply the Midpoint Rule

Finally, we apply the Midpoint Rule to approximate the integral as follows: \(\int_{0}^{1} \sin(\pi x) d x \approx \Delta x \sum_{i=0}^{n-1} f(x_i)\) \(\approx \frac{1}{6}\left[\sin\left(\pi \cdot \frac{1}{12}\right) + \sin\left(\pi \cdot \frac{1}{4}\right) + \sin\left(\pi \cdot \frac{5}{12}\right) + \sin\left(\pi \cdot \frac{7}{12}\right) + \sin\left(\pi \cdot \frac{3}{4}\right) + \sin\left(\pi \cdot \frac{11}{12}\right)\right]\) Now, we can either leave the answer as it is, or evaluate the numerical value using a calculator.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Numerical Integration
Numerical integration is a mathematical technique used to approximate the value of a definite integral when an exact solution is difficult or impossible to determine. It is particularly useful when dealing with functions that do not have simple antiderivatives or when we're working with empirical data points.

Numerical methods, including the Midpoint Rule, Trapezoidal Rule, and Simpson's Rule, transform the problem of finding the area under a curve into a problem of summing areas of simpler shapes like rectangles or trapezoids. Among these methods, the Midpoint Rule is known for its simplicity and reasonable precision for a certain class of functions.

The exercise provided demonstrates how to apply the Midpoint Rule to estimate the integral of the function \(\sin(\pi x)\) over the interval [0, 1]. This method uses the midpoints of subintervals to calculate the height of rectangles whose combined areas approximate the area under the curve.
Riemann Sums
Riemann sums are a way to approximate the definite integral of a function using finite sums. They are the foundation of numerical integration methods and integral calculus. A Riemann sum comes in four main flavors depending on where the sample point is chosen: left endpoint, right endpoint, midpoint, and a random point within each subinterval.

The Midpoint Rule, which the exercise uses, is a specific type of Riemann sum that takes the function's value at the midpoint of each subinterval to represent the height of the rectangle. The choice of the midpoint can offer a better approximation than using the left or right endpoints because it often balances out the overestimation and underestimation of the area under the curve.

To improve understanding, it's helpful to visualize this geometrically by sketching the function and the rectangles that the Midpoint Rule creates. Seeing this visual representation can clarify how the Riemann sum approximates the integral.
Definite Integral
The definite integral of a function represents the net area under the curve defined by the function over a certain interval. It has wide-ranging applications in physics, engineering, and economics, where it can be used to calculate quantities like displacement, work, and total profit.

In the given exercise, the definite integral \(\int_{0}^{1} \sin(\pi x) dx\) is being approximated. The exact area under the sine curve from 0 to 1 is sought, but due to the complexity of the sine function's antiderivative and the presence of \(\pi\), it's preferable to use a numerical method like the Midpoint Rule. This rule provides a practical way to estimate definite integrals with a manageable margin of error, especially as the number of subintervals increases, leading to a more precise result.

By following the solution's steps, one can learn how to approach similar problems. The step by step process includes dividing the interval, calculating midpoints, evaluating the function at these points, and then summing up the areas of the rectangles created to get an approximation of the definite integral.

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Most popular questions from this chapter

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