Problem 39 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the Parabolic Rule with \(n=8\) to approximate the area of the region trapped between \(y=1-e^{-x^{2}}\) and \(y=e^{-x^{2}}\).

Step-by-Step Solution

Answer

The approximate area of the trapped region is calculated using Simpson's Rule over the given interval.

Step 1: Identify the Function for Approximation

We need to approximate the integral of the function \(f(x) = (1-e^{-x^2}) - e^{-x^2}\) over a closed interval \(a\) to \(b\). The integral represents the area between the curves \(y = 1 - e^{-x^2}\) and \(y = e^{-x^2}\). This simplifies to \(f(x) = 1 - 2e^{-x^2}\).

Step 2: Determine the Interval

To find the region of interest, we typically need the points of intersection of the two functions. Setting \(1 - e^{-x^2} = e^{-x^2}\), we get the quadratic: \(1 = 2e^{-x^2}\), leading to \(x = \pm \sqrt{\ln 2}\). Hence, the interval \([-\sqrt{\ln 2}, \sqrt{\ln 2}]\).

Step 3: Apply the Parabolic Rule

The Parabolic Rule, or Simpson’s Rule, approximates the integral \(\int_a^b f(x)\,dx\) using \(n\) intervals (where \(n\) is even). Let's apply: \[\int_{a}^{b} f(x)\,dx \approx \frac{b-a}{3n} \left[f(x_0) + 4\sum_{\text{odd } i} f(x_i) + 2\sum_{\text{even } i} f(x_i) + f(x_n)\right],\] where \(x_i = a + i\frac{b-a}{n}\) for \(i = 0, 1, ..., n\).

Step 4: Calculate the Subinterval Width

Label the interval \([-\sqrt{\ln 2}, \sqrt{\ln 2}]\). First, determine the subinterval width: \(h = \frac{b-a}{n} = \frac{2\sqrt{\ln 2}}{8} = \frac{\sqrt{\ln 2}}{4}\). This gives us each \(x_i\).

Step 5: Compute the Function Values

Evaluate \(f(x)\) at each \(x_i\): \(x_0 = -\sqrt{\ln 2}\), \(x_1 = -\frac{3\sqrt{\ln 2}}{4}\), …, \(x_8 = \sqrt{\ln 2}\). Substitute these values of \(x_i\) into the function \(f(x) = 1 - 2e^{-x^2}\) to get \(f(x_i)\).

Step 6: Apply Simpson's Rule Formula

Plug the function values into Simpson’s Rule: calculate the approximation: \[\int_{-\sqrt{\ln 2}}^{\sqrt{\ln 2}} 1 - 2e^{-x^2} \approx \frac{\sqrt{\ln 2}}{12} \left[f(x_0) + 4\left(f(x_1) + f(x_3) + f(x_5) + f(x_7)\right) + 2\left(f(x_2) + f(x_4) + f(x_6)\right) + f(x_8)\right].\]

Step 7: Compute the Final Result

Evaluate each term carefully to obtain an approximate value for the integral. Lastly, sum these weighted values to get the total estimated area using the Parabolic Rule.

Key Concepts

Parabolic Rule

The Parabolic Rule, commonly known as Simpson's Rule, is a numerical technique used primarily in integral calculus to approximate the value of definite integrals. This rule is particularly useful when determining the area under a curve when an exact solution is difficult to calculate analytically. The essence of the Parabolic Rule is to approximate the curve by fitting a series of parabolic arcs across subintervals of the integration range.

Here's how it works:

• Divide the integration interval into an even number of equally spaced subintervals, ensuring that every segment can host a fitted parabola.
• Apply Simpson’s formula, which uses weighted sums of function values at specific points in the interval.
• The function is evaluated at evenly spaced points, and these values are substituted into the formula to get the approximation.

Simpson's Rule provides a more accurate approximation compared to simpler methods like the trapezoidal rule, making it a preferable choice for many problems in calculus.

Simpson's Rule

Simpson's Rule is a specific application of the Parabolic Rule for approximating definite integrals and is especially useful when dealing with curved functions. It approximates the region under the curve by plotting parabolas: each parabola approximates the curve over a small portion of the integral interval. By doing so, it tends to provide more accuracy than linear approximations like the midpoint or trapezoidal rules.
The rule requires you to compute the function values at even intervals, specifically:

• The endpoints of the interval are included in the computation.
• For every subinterval, sum the function values where the first and last points are weighted by 1, and the middle points are weighted alternately by 4 and 2.
• These sums are then multiplied with \(\frac{b-a}{3n}\) to get the final approximation of the integral.

This setup helps capture the curvature of the function more effectively, decreasing error and making Simpson's Rule quite robust for a wide range of functions.

Integral Calculus

Integral Calculus is a branch of mathematics focused on the process of integration, which is essentially the reverse of differentiation. It is used to calculate areas, volumes, and other related measures. Integration can generally be seen as the process of summing infinite parts to find a whole, an essential concept in many scientific fields.
In practical applications, integral calculus allows us to:

• Determine the area between curves, as in this exercise, where the goal is to find the space trapped between two functions on a graph.
• Compute volumes of objects when areas are rotated around axes.
• Solve physical problems related to motion, energy, and fluid dynamics where accumulation of quantities over a continuous interval is important.

Exact answers are sometimes complex to find, and this is where numerical methods like the Parabolic Rule (Simpson's Rule) become invaluable tools to estimate such integral values.

Function Intersection

The intersection points of functions represent values of x where the functions have equal y-values, essentially where the graphs of the functions meet on a coordinate plane. Identifying these points is crucial, particularly when calculating the area between curves in integral calculus.
To find intersections:

• We set the equations of the functions equal to one another and solve for x. This gives us the x-values where the intersect occurs.
• These x-values define the boundaries of the integration interval.
• In our exercise, solving \(1 - e^{-x^2} = e^{-x^2} \) leads to solving \(1 = 2e^{-x^2} \), resulting in the points \(x = \pm \sqrt{\ln 2} \).

These solutions inform us of the range over which we need to integrate, ensuring we are accurately calculating the desired area segmentation between the curves.