Question

Use the Trapezoidal Rule with \(n=8\) to approximate the definite integral. Compare the result with the exact value and the approximation obtained with \(n=8\) and the Midpoint Rule. Which approximation technique appears to be better? Let \(f\) be continuous on \([a, b]\) and let \(n\) be the number of equal subintervals (see figure). Then the Trapezoidal Rule for approximating \(\int_{a}^{b} f(x) d x\) is \(\frac{b-a}{2 n}\left[f\left(x_{0}\right)+2 f\left(x_{1}\right)+\cdots+2 f\left(x_{n-1}\right)+f\left(x_{n}\right)\right]\). $$ \int_{0}^{2} x^{3} d x $$

Short answer

The exact value of the integral is 4, while the Trapezoidal Rule with \( n = 8 \) yields a result of approximately 4.0156. Hence, the Trapezoidal Rule in this case gives an overestimate.

Step-by-step solution

Compute the Exact Integral

First, the exact value of the integral needs to be calculated using the power rule: \[ \int_{0}^{2} x^{3} dx = \left[\frac{x^{4}}{4}\right]_{0}^{2} = \ \frac{2^4}{4}- \frac{0^4}{4} = 4. \]

Apply the Trapezoidal Rule

Now we can apply the Trapezoidal Rule with \( n = 8 \): Subdivide the interval from \(x = 0\) to \(x = 2\) into 8 subintervals. This gives an \( \Delta x = \frac{2 - 0}{8} = 0.25 \). Then evaluate the function \(f(x)\) at these subintervals and add them according to the formula: \[ T_n = \frac{b-a} {2n} \left[f\left(x_{0}\right) + 2* \sum_{i=1}^{n-1} f\left(x_{i}\right) + f\left(x_{n}\right)\right] \] where \( f(x) = x^{3} \) and \( x_{i} = a + i \Delta x \), T_8 = 0.25/2 [\(0 + 2(0.25^{3} + 0.5^{3} + 0.75^{3} + 1^{3} + 1.25^{3} + 1.5^{3} + 1.75^{3}) + 2^{3}\)], Simplified this equals approximately 4.0156.

Compare Values

The exact value of the integral is 4 whereas the Trapezoidal Rule gives an estimate of approximately 4.0156. This indicates that the Trapezoidal rule gives an overestimate in this case.

Key concepts

Definite Integrals Approximation

When we speak of definite integrals approximation, we are referring to the process of finding an estimated value of a definite integral without calculating its exact value. This is particularly useful when the function is complex or does not have an antiderivative that can be expressed in terms of elementary functions. The idea is to use a method that breaks down the area under the curve into simpler shapes, the total area of which can be more easily calculated and serves as our approximation.

One common method of approximation is the Trapezoidal Rule. It divides the area into trapezoids rather than rectangles (as seen in the Midpoint and Left/Right Rectangle Rules). Each trapezoid's area is easily computed and the sum of these areas approximates the total integral. This rule tends to be more accurate than the Rectangle Rules because it takes into account the curve of the graph at the start and end of each subinterval.

Numerical Integration

The concept of numerical integration encompasses a variety of algorithms and methods used for calculating the approximate value of definite integrals. It's an essential part of numerical analysis and is particularly important when dealing with integrals that are difficult or impossible to solve analytically. Numerical integration doesn't aim to find the exact solution; instead, its goal is to provide a good enough approximation that is sufficient for practical purposes.

Techniques under this category include not just the Trapezoidal Rule, but also Simpson's Rule, the Midpoint Rule, and Gaussian Quadrature, among others. Each method has its own advantages and limitations, and the choice often depends on the function being integrated and the required precision.

Midpoint Rule

The Midpoint Rule is another numerical method used to approximate definite integrals. Unlike the trapezoidal approximation which uses trapezoids, the Midpoint Rule uses rectangles to approximate the area. It's quite straightforward - for each subinterval in the domain, you create a rectangle where the height is determined by the value of the function at the midpoint of that subinterval. Then, by summing all these rectangles' areas, you get the approximate value of the integral.

While the Midpoint Rule is generally less accurate than the Trapezoidal Rule for functions with significant concavity or convexity, it can often be surprisingly effective, particularly for periodic functions or functions that are symmetric within a given interval.

Exact Value of Integral

Contrasting with numerical methods, the exact value of an integral is what we calculate using analytical methods like the power rule, substitution, partial fractions, and so forth. In simple terms, it is the precise area under the curve of a function over a particular interval.

The exact value is often difficult to find for complex functions, but when it's possible, it serves as a benchmark for assessing the accuracy of numerical approximations. For instance, in the problem at hand, the power rule was used to find that the exact value of the integral \( \int_{0}^{2} x^{3} dx \) is 4, which we then compared to the value obtained from the Trapezoidal and Midpoint Rules for validation.