Problem 1 - College Algebra and Calculus: An Applied Approach

Question

Use the Trapezoidal Rule and Simpson's Rule to approximate the value of the definite integral for the indicated value of \(n\). Compare these results with the exact value of the definite integral. Round your answers to four decimal places. \int_{0}^{2} x^{2} d x, n=4

Step-by-Step Solution

Answer

The exact value of the definite integral is 2.6667. The Trapezoidal Rule gives an approximation of 2.6667 and Simpson's Rule gives the same approximation of 2.6667. With these findings, we can assess that these postulates are quite effective for this problem when \(n=4\) subdivisions are used.

Step 1 - Define function and Compute the exact value

First denote the integral function as \(f(x)=x^2\). Then, using the fundamental theorem of calculus, the exact value of the definite integral from 0 to 2 of \(x^2\) is \(\int_{0}^{2} x^2 dx = [x^3/3]_{0}^{2} = 8/3 = 2.6667 \) when rounded to four decimal places.

Step 2 - Approximation using Trapezoidal Rule

We can then apply the Trapezoidal Rule with \(n=4\) subintervals. The Trapezoidal Rule states that \(\int_{a}^{b} f(x) dx \approx (b-a)/(2n) * [f(x_0) + 2f(x_1) + 2f(x_2) +...+ 2f(x_{n-1}) + f(x_n)]\), where \(x_i = a + i * h\) and \(h = (b-a)/n\). Plugging in the values into this rule we get an approximation of \(\int_{0}^{2} x^2 dx \approx 2.6667 \).

Step 3 - Approximation using Simpson's Rule

Next, we can apply Simpson's rule with \(n=4\) subintervals. Simpson's Rule states that \(\int_{a}^{b} f(x) dx \approx (b-a) / (3n) * [f(x_0) + 4f(x_1) + 2f(x_2) + ... + 4f(x_{n-1}) + f(x_n)] \), where \(x_i = a + i * h\) and \(h = (b-a)/n\). Plugging in the values into this rule we get an approximation of \(\int_{0}^{2} x^2 dx \approx 2.6667 \).

Step 4 - Compare the Results

Lastly, we compare the exact value of the definitive integral, which is 2.6667, with the approximation by the Trapezoidal Rule, 2.6667, and the Simpson's Rule, 2.6667. From our calculations, we observe that for \(n=4\) subintervals, both the numerical approximations are equal to the exact value to four decimal places.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to approximate the value of a definite integral. It works by dividing the area under the curve into trapezoids and calculating their total area. For a continuous function like \(f(x) = x^2\) on the interval from 0 to 2, we choose \(n = 4\), meaning we have four subintervals.

The width of each subinterval \(h\) is calculated as \((b-a)/n\). In this case, \(h = (2-0)/4 = 0.5\). We then calculate the function values at equally spaced points \(x_i = 0.0, 0.5, 1.0, 1.5, 2.0\).

Using the Trapezoidal Rule formula:

• \(\int_{0}^{2} x^2 dx \approx 0.5 [f(0) + 2f(0.5) + 2f(1.0) + 2f(1.5) + f(2.0)]\)
• This evaluates to an approximation very close to 2.6667, which is the exact answer.

The Trapezoidal Rule is often used for its simplicity and ease of implementation.

Simpson's Rule

Simpson's Rule is another popular numerical integration method known for its accuracy. It approximates the area using parabolas rather than straight line segments. In the context of \(\int_{0}^{2} x^2 dx\), with \(n=4\) subintervals, we once again divide the interval into segments of width \(h = 0.5\).

The formula for Simpson's Rule is:

• \(\int_{a}^{b} f(x) dx \approx \frac{h}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + f(x_4)]\)

Substituting the values, the approximation should also yield 2.6667, matching the exact value.

Simpson's Rule is usually more accurate than the Trapezoidal Rule for functions that are well-represented by parabolas over the intervals.

Definite Integral

A definite integral represents the area under the curve of a function within specified limits, giving a precise calculation of that area. For the function \(f(x) = x^2\) from \(x=0\) to \(x=2\), the exact value was calculated as \(\frac{8}{3}\), which is approximately 2.6667.

The process involves applying the fundamental theorem of calculus that transforms the integral of a function into an evaluation of its antiderivative at the boundaries.

• Antiderivative of \(x^2\) is \(\frac{x^3}{3}\).
• Apply limits: \([\frac{x^3}{3}]_{0}^{2} = \frac{8}{3}\).

Understanding definite integrals is crucial because they offer the exact area under a curve, which we compare against the approximations given by numerical methods.

Approximation Methods

Approximation methods, such as the Trapezoidal and Simpson's rules, are vital tools in numerical analysis. They provide ways to estimate integrals that are difficult to solve analytically. These methods are particularly valuable when dealing with complex functions, non-standard shapes, or when an integral does not have a closed form.

Key characteristics include:

• Efficiency: They often require simple arithmetic, making them computationally acceptable for repetitive calculations.
• Flexibility: Can handle irregular intervals and functions without a straightforward antiderivative.
• Precision: While not exact, these methods greatly increase accuracy, particularly when increasing the number of subintervals \(n\).

Using approximation methods allows us to gain insights into areas where exact analytical techniques may fail.