Question

Use the Parabolic Rule with \(n=8\) to approximate the area of the region trapped between \(y=\ln (x+1)\) and \(y=x / 4\). Hint: One point of intersection is obvious; the other you must approximate.

Short answer

The approximate area is 0.847.

Step-by-step solution

Find Points of Intersection

The points of intersection are where the functions are equal: \( \ln(x+1) = \frac{x}{4} \). The first intersection is at \(x = 0\) since \(\ln(1) = 0\). To find the other intersection, solve \(\ln(x+1) \approx \frac{x}{4}\) using numerical methods. This gives the approximate intersection at \(x \approx 1.84\).

Define the Interval and Calculate Step Size

The interval for integration is between the two points of intersection, \([0, 1.84]\). With \(n = 8\), calculate the step size \(h = \frac{1.84 - 0}{8} = 0.23\).

Setting Up the Parabolic Rule (Simpson's Rule)

Simpson's Rule for \(n=8\) gives: \[ \text{Approximate Area} = \frac{h}{3} [f(x_0) + 4(f(x_1) + f(x_3) + f(x_5) + f(x_7)) + 2(f(x_2) + f(x_4) + f(x_6)) + f(x_8)] \].

Calculate Function Values

The function values are calculated at the points: \(x_0 = 0\), \(x_1 = 0.23\), \(x_2 = 0.46\), \(x_3 = 0.69\), \(x_4 = 0.92\), \(x_5 = 1.15\), \(x_6 = 1.38\), \(x_7 = 1.61\), and \(x_8 = 1.84\). Calculate \(f(x) = \ln(x+1) - \frac{x}{4}\) for each, obtaining: \(f(0) = 0\), \(f(0.23) \approx 0.191\), \(f(0.46) \approx 0.345\), \(f(0.69) \approx 0.467\), \(f(0.92) \approx 0.561\), \(f(1.15) \approx 0.636\), \(f(1.38) \approx 0.696\), \(f(1.61) \approx 0.742\), \(f(1.84) \approx 0.776\).

Plug Values into Simpson's Rule

Now substitute these into Simpson's Rule: \[\text{Approximate Area} = \frac{0.23}{3} [0 + 4(0.191 + 0.467 + 0.636 + 0.742) + 2(0.345 + 0.561 + 0.696) + 0.776] \].

Calculate the Final Result

Calculate the expression: \[\frac{0.23}{3} [0 + 4(2.036) + 2(1.602) + 0.776] \]. Simplifying gives: \[\frac{0.23}{3} \times 11.048 \approx 0.847\].

Key concepts

Numerical Integration

Numerical integration is a technique to approximate the value of an integral, especially when it is challenging to find the exact solution. This process is crucial for functions that do not have simple antiderivatives or definite integrals that are difficult to compute analytically. In simple terms, it allows us to calculate the area under a curve represented by a complex function. This is often necessary in real-world applications where the exact integral cannot be obtained easily.

One of the most popular methods for numerical integration is Simpson's Rule. This method uses parabolic arcs instead of straight lines to approximate the function, providing more accurate results compared to other techniques like the Trapezoidal Rule. In the exercise at hand, Simpson's Rule is employed to approximate the area between two curves, namely, a logarithmic function and a linear function.

• Simpson's Rule works best with an even number of intervals, making its application straightforward and effective for the exercise.
• The method involves splitting the integration interval into multiple sub-intervals of equal width and fitting a quadratic polynomial over each pair of consecutive sub-intervals.

Area Approximation

Area approximation is the task of estimating the space enclosed by a curve or between multiple curves. This computation is essential in fields like physics and engineering, where it helps calculate quantities like work done by a force or the volume of a solid of revolution.

In the provided exercise, the area between the curves of the logarithmic function, \(y = \ln(x+1)\), and the linear function, \(y = \frac{x}{4}\), is approximated using Simpson's Rule. The integration limits are the points where the two functions intersect, which are found to be approximately at \(x = 0\) and \(x \approx 1.84\).

• These points serve as the boundaries for the integration interval, thus defining the region over which the area is calculated.
• Simpson's Rule is beneficial in this context as it considers more function values than the Trapezoidal Rule, offering a more precise approximation of the area.

Points of Intersection

Points of intersection involve finding where two or more functions share the same value for both the x and y coordinates. This concept is critical in various mathematical calculations such as finding solution sets for equations and determining boundaries for integration.

For the given problem, the intersection points are where the functions \(y = \ln(x+1)\) and \(y = \frac{x}{4}\) equal each other. These are solved by setting the functions equal to each other, resulting in the equation \(\ln(x+1) = \frac{x}{4}\).

• The obvious point of intersection is at \(x = 0\), because \(\ln(1) = 0\).
• The other point is found to be approximately \(x \approx 1.84\) using numerical or graphical methods, as the equation cannot be solved easily by analytical means.

These points define the limits for the numerical integration using Simpson's Rule.

Function Values Calculation

Function values calculation is an essential step in applying Simpson's Rule. It involves computing the function’s value at specific points, which are determined by dividing the interval of interest into smaller sub-intervals based on the step size. This step is crucial because these calculated values are used to approximate the area under the curve.

In the case of this exercise, we calculate the function values at nine equally spaced points between the two intersection points \(x = 0\) and \(x = 1.84\). The step size is determined as \(h = \frac{1.84 - 0}{8} = 0.23\).

• For each of these points, the function \(f(x) = \ln(x+1) - \frac{x}{4}\) is evaluated, yielding values like \(f(0) = 0\), \(f(0.23) \approx 0.191\), and so on.
• These calculated values are then substituted into Simpson's formula, ensuring an accurate estimation of the area between the curves.