Problem 10 - 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. $$ \int_{0}^{2} \sqrt{1+x} d x, n=4 $$

Step-by-Step Solution

Answer

Using the Trapezoidal Rule, the approximation of the integral is 2.4747. Using Simpson's Rule, the approximation is 2.5214. The exact value of the integral is 2.3852.

Step 1: Trapezoidal Rule

Using the trapezoidal rule with \(n=4\) gives \(h=(b-a)/n=(2-0)/4=0.5\). The trapezoidal approximation is thus given by \(T_{n}=h/2[f(a)+2\sum_{j=1}^{n-1}f(a+jh)+f(b)]\). Plugging in the relevant values, we have \(T_{4}=0.5/2[\sqrt{1+0}+2(\sqrt{1+0.5}+\sqrt{1+1}+\sqrt{1+1.5})+\sqrt{1+2}]\), which equals 2.4747 when evaluated.

Step 2: Simpson's Rule

Next, Simpson's Rule with \(n=4\) results in \(h=(b-a)/n=(2-0)/4=0.5\). The Simpson's Rule approximation is \(S_{n}=h/3[f(a)+4\sum_{j=1,3,5,\dots}^{n-1}f(a+jh)+2\sum_{j=2,4,6,\dots}^{n-2}f(a+jh)+f(b)]\). Substituting the appropriate values, we get \(S_{4}=0.5/3[\sqrt{1+0}+4(\sqrt{1+0.5}+\sqrt{1+1.5})+2(\sqrt{1+1})+\sqrt{1+2}]\), which equals 2.5214 when calculated.

Step 3: Comparing to Exact Value

The exact value of the integral can be calculated by computing the antiderivative: \(\frac{2}{3}(2\sqrt{3}-1)=2.3852\). Comparing this to the approximations made with the trapezoidal rule and Simpson's Rule reveals that both methods provide good approximations, with the Simpson's Rule approximation being especially close to the exact value.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to approximate the value of a definite integral. Essentially, this method works by dividing the area under a curve into a series of trapezoids, rather than rectangles, to better fit the shape of the curve. This approach can provide more accurate results than more basic methods when calculating an integral numerically.

The formula for the Trapezoidal Rule is given by:

• \( T_{n} = \frac{h}{2} \left[ f(a) + 2 \sum_{j=1}^{n-1} f(a + jh) + f(b) \right] \)

Where:

• \( h \) is the width of each trapezoid, calculated as \( \frac{b-a}{n} \)
• \( f(a) \) and \( f(b) \) are the values of the function at the endpoints of the integral

This rule assumes that the function can be approximated by straight lines between these points.

In our original exercise, the integral \( \int_{0}^{2} \sqrt{1+x} \, dx \) was approximated using 4 trapezoids (\( n=4 \)), resulting in an estimated value of 2.4747. This gives us a sense of the area under the curve from 0 to 2, calculated by evaluating the function at specific points and averaging these results.

Simpson's Rule

Simpson's Rule is another numerical method designed to approximate definite integrals. It differs from the Trapezoidal Rule by using parabolic arcs instead of straight-line segments to approximate the curve. This can make it more accurate, especially for smooth functions and symmetric intervals.

The Simpson's Rule formula is expressed as:

• \( S_{n} = \frac{h}{3} \left[ f(a) + 4 \sum_{j=1,3,5,\ldots}^{n-1} f(a + jh) + 2 \sum_{j=2,4,6,\ldots}^{n-2} f(a + jh) + f(b) \right] \)

With Simpson's Rule, the function's behavior is closely tracked by quadratics. This provides an approximation that can be much closer to the true integral than linear approximations.

In our calculation of the original exercise \( \int_{0}^{2} \sqrt{1+x} \, dx \) with 4 subintervals, Simpson's Rule gave a value of 2.5214, which was closer to the exact value. Its higher accuracy is owed to the way it better captures changes in the function's slope and curvature across its interval.

Definite Integral

A definite integral represents the signed area under a curve, bounded between two limits, usually denoted as \( a \) to \( b \). This concept is foundational in calculus and is often used to compute areas, volumes, and other quantities for which changes occur over an interval.

The process of finding a definite integral involves calculating the antiderivative, or the indefinite integral, of a function, then applying the limits of integration. Mathematically, this is represented by the formula:

• \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \)

Where \( F(x) \) is the antiderivative of \( f(x) \).

In our example, we dealt with the integral \( \int_{0}^{2} \sqrt{1+x} \, dx \). Calculating the antiderivative gives us \( \frac{2}{3}(2\sqrt{3}-1) \), which evaluates to 2.3852. This exact solution showcases how the area under the curve between the points 0 and 2 is accounted for precisely by analyzing the change of the function across that interval.