Question

Approximate the definite integral with the Trapezoidal Rule and Simpson's Rule, with \(n=6\). $$ \int_{1}^{4} \ln x d x $$

Short answer

The Trapezoidal Rule and Simpson's Rule approximations are obtained by applying their respective formulas to the calculated values.

Step-by-step solution

Understand the Problem

We need to approximate the definite integral \( \int_{1}^{4} \ln x \, dx \) using both the Trapezoidal Rule and Simpson's Rule with \( n=6 \). The interval of integration \([1, 4]\) will be divided into 6 equal subintervals.

Calculate the Interval Width

The interval width \( h \) is calculated as follows:\[ h = \frac{b-a}{n} = \frac{4-1}{6} = \frac{1}{2} \] where \( a=1 \) and \( b=4 \).

Determine Endpoint Values

We calculate the x-values at the endpoints of each subinterval as follows: - \( x_0 = 1 \)- \( x_1 = 1.5 \)- \( x_2 = 2 \)- \( x_3 = 2.5 \)- \( x_4 = 3 \)- \( x_5 = 3.5 \)- \( x_6 = 4 \)

Calculate Function Values

Evaluate the function \( f(x) = \ln x \) at each endpoint:- \( f(x_0) = \ln 1 = 0 \)- \( f(x_1) = \ln 1.5 \)- \( f(x_2) = \ln 2 \)- \( f(x_3) = \ln 2.5 \)- \( f(x_4) = \ln 3 \)- \( f(x_5) = \ln 3.5 \)- \( f(x_6) = \ln 4 \)

Apply the Trapezoidal Rule Formula

Using the Trapezoidal Rule formula:\[\text{Trapezoidal Rule} = \frac{h}{2} [f(x_0) + 2(f(x_1) + f(x_2) + f(x_3) + f(x_4) + f(x_5)) + f(x_6)]\]Substitute the values into the formula and calculate.

Apply Simpson's Rule Formula

Simpson's Rule requires an even number of subintervals, which we have (\( n=6 \)). The formula is:\[\text{Simpson's Rule} = \frac{h}{3} [f(x_0) + 4(f(x_1) + f(x_3) + f(x_5)) + 2(f(x_2) + f(x_4)) + f(x_6)]\]Substitute the calculated function values into the formula to get the approximation.

Key concepts

Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to approximate the value of definite integrals. This approach works by dividing the area under the curve into a series of trapezoids rather than attempting to find an exact integral when such is cumbersome or impossible analytically. Each trapezoid approximates the area of the section of the curve between the subintervals. To apply this method, you first need to determine the interval width, which is done using the formula:\[ h = \frac{b-a}{n} \]where \(a\) and \(b\) are the limits of integration, and \(n\) is the number of subintervals. For instance, when calculating \( \int_{1}^{4} \ln x \, dx \) using 6 subintervals, we get \( h = \frac{1}{2} \). The next step involves calculating the function's values \( f(x) \) at each interval point. Then, plug these values into the Trapezoidal Rule formula:\[ \text{Trapezoidal Rule} = \frac{h}{2} [f(x_0) + 2(f(x_1) + f(x_2) + f(x_3) + f(x_4) + f(x_5)) + f(x_6)] \] The trapezoid's approximate "height" averages out the curve's areas, providing a straightforward means to estimate the integral. This rule is particularly useful for functions that are linear or near linear over intervals, offering a quick approximation.

Simpson's Rule

Simpson's Rule is another popular method for approximating definite integrals and is generally more accurate than the Trapezoidal Rule, especially for polynomial functions. Unlike the Trapezoidal Rule, which treats each segment as a linear trapezoid, Simpson's Rule uses parabolic arcs to approximate the area under the curve. The basic idea is to fit a quadratic polynomial between every pair of subintervals. Hence, the curve segments would closely reflect the function's behavior over the intervals, rather than straight lines.To apply Simpson's rule, ensure the number of subintervals \(n\) is even. Following the same initial steps of calculating subinterval values and function values, the Simpson's Rule formula can be applied:\[ \text{Simpson's Rule} = \frac{h}{3} [f(x_0) + 4(f(x_1) + f(x_3) + f(x_5)) + 2(f(x_2) + f(x_4)) + f(x_6)] \] This use of multiples — 4 for odd-indexed terms and 2 for even-indexed ones, along with one for each endpoint — ensures the parabola fits closely enhances accuracy. Simpson's Rule tends to provide better approximations when dealing with smooth curves or integrations over balanced intervals.

Definite Integral

The concept of a definite integral involves finding the total area under a curve between two specific bounds \([a, b]\). Not only does it provide an accumulation of all values or changes of a given function over that interval, but it is also fundamental in many disciplines. In mathematics and physics, definite integrals help calculate areas, volumes, displacement, and accumulated quantities. They are written in the form:\[ \int_{a}^{b} f(x) \, dx \]where \(a\) and \(b\) are the lower and upper limits of the integral, respectively.For functions that resist straightforward antiderivatives or when quick approximations are needed, numerical methods such as the Trapezoidal and Simpson's Rules come in handy. These approximation techniques permit the consistent estimation of definite integrals for complex or non-standardized functions.Understanding the foundation of definite integrals gives insight into a function's full behavior over an interval, contributing to a wide range of applications like analyzing trends, optimization problems, and solving real-world issues involving rates of change and accumulation.