Question

Comparing the Midpoint and Trapezoid Rules Compare the errors in the Midpoint and Trapezoid Rules with \(n=4,8,16,\) and 32 subintervals when they are applied to the following integrals (with their exact values given). $$\int_{0}^{1}\left(8 x^{7}-7 x^{8}\right) d x=\frac{2}{9}$$

Short answer

Question: Compare the errors of the Midpoint and Trapezoid Rules for approximating the definite integral of the function $$f(x) = 8x^7 - 7x^8$$ on the interval $$[0, 1]$$ for $$n=4,8,16,32$$ subintervals, considering that the exact value of the integral is $$\frac{2}{9}$$.

Step-by-step solution

Define the integral and given information

The definite integral we are working on is $$\int_{0}^{1}\left(8 x^{7}-7 x^{8}\right) d x=\frac{2}{9}$$ The function to integrate: $$f(x) = 8x^7 - 7x^8$$ The interval of integration: $$[0, 1]$$ And we will use the Midpoint and Trapezoid Rules with 4, 8, 16, and 32 subintervals.

Calculate the step size for each value of n

For each provided value of n (4, 8, 16, and 32), we will calculate the step size using the formula: $$\Delta x = \frac{b-a}{n}$$ Where $$a$$ and $$b$$ are the lower and upper limits of integration, respectively, and $$n$$ is the number of subintervals.

Implement the Midpoint and Trapezoid Rules

For each value of n, apply the Midpoint and Trapezoid Rules: Midpoint Rule: $$ M_{n} = \Delta x \sum_{i=1}^n f\left(\frac{x_{i-1}+x_i}{2}\right) $$ Trapezoid Rule: $$ T_{n} = \frac{\Delta x}{2}\left(f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b)\right) $$

Calculate the errors and compare the results

For each value of n, estimate the errors using the Midpoint and Trapezoid Rules. The error can be calculated using the formula: $$error = |exact\_value - approximation|$$ Compare the errors for each rule with different values of n.

Final Output

For each n (4, 8, 16, 32), perform steps 2–4 to compute the step size, apply the Midpoint and Trapezoid Rules, calculate the errors, and compare their results. This will help you understand which method performs better for different choices of n.

Key concepts

Midpoint Rule

The Midpoint Rule is a numerical method used to estimate the value of a definite integral. This rule approximates the area under a curve by considering the height of the function at the midpoint of each subinterval. To apply this tool, we divide the interval of integration into a specific number of equally sized parts, called subintervals. Then, we calculate the function's value at the midpoint of each subinterval, and multiply it by the width of these subintervals to obtain the area of rectangles that approximate the curve.

Mathematically, for a function \( f(x) \) and interval \( [a, b] \) divided into \( n \) subintervals, the approximation is given by:
\[ M_n = \Delta x \sum_{i=1}^n f\left(\frac{x_{i-1}+x_i}{2}\right) \]
where \( \Delta x \) is the width of each subinterval, and \( x_{i-1} \) and \( x_i \) are the endpoints of the i-th interval. The midpoint is used as it often provides a better approximation than simply using the endpoints.

Trapezoid Rule

The Trapezoid Rule is another method for approximating the value of a definite integral, but instead of rectangles, it uses trapezoids to model the area under a curve. By doing this, the approximation can take into account the slope of the function within each subinterval, which often leads to a more accurate estimate compared to the Midpoint Rule, especially when the function is approximately linear over each subinterval.

The formula for the Trapezoid Rule is:
\[ T_{n} = \frac{\Delta x}{2}(f(a) + 2\sum_{i=1}^{n-1} f(x_i) + f(b)) \]
In this equation, \( \Delta x \) is the width of subintervals, \( f(a) \) and \( f(b) \) are the values of the function at the start and end points, and the sum calculates the contribution of the function's value at each interior point. The factors of 2 account for the fact that each interior point is shared by two trapezoids.

Definite Integral

A definite integral provides the exact area under a curve bounded by two points, typically referred to as the limits of integration. The integral \( \int_{a}^{b} f(x) dx \) represents the accumulation of the function's values from \( x = a \) to \( x = b \). This concept is crucial because it encapsulates not just the notion of area under a function's graph, but also various physical meanings, such as displacement given velocity over a time interval, or the total work done by a variable force.

The exact value of a definite integral is often found using antiderivatives when the function is integrable in simple forms, but for more complex cases or when a quick approximation is needed, numerical methods like the Midpoint and Trapezoid Rules come into play.

Numerical Approximation Error

An inherent issue with using numerical methods like the Midpoint and Trapezoid Rules is that they yield approximations, which means there's always some level of numerical approximation error. This error represents the difference between the exact value of the definite integral and the estimated value produced by the numerical method.

To calculate this error, we use the formula:
\[error = |exact\_value - approximation|\]
This not only allows us to measure the accuracy of our approximation but also to compare different numerical methods. Generally, the more subintervals used (larger \( n \)), the smaller the error, as the shapes used in the approximation more closely resemble the actual area under the curve. However, it's important to note that reducing the error to an acceptable level depends on the behavior of the function over the interval and the specific numerical method used.