Problem 14 - 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. $$ \int_{0}^{2} x \sqrt{x^{2}+1} d x, n=4 $$

Step-by-Step Solution

Answer

The full work to compute these values might extend beyond this general framework, but what's important are these four steps: approximating via the Trapezoidal rule, approximating via Simpson's rule, calculating the exact value of the integral, and comparing the three values.

Step 1: Trapezoidal Rule

Firstly, the Trapezoidal rule is used to approximate the value of the integral. The general formula for Trapezoidal rule is: \[T_n=\frac{h}{2} [f(a) + 2 \sum_{i=1}^{n-1} f(a+ih)+f(b)]\] where \(h = (b-a)/n\) is the width of each of the \(n\) subintervals of [a, b]. The values of \(a\) and \(b\) from the given integral are 0 and 2, respectively, and \(n=4\). To find \(T_4\), substitute these values into the formula with the given function \(f(x) = x \sqrt{x^{2}+1}\) to obtain the Trapezoidal Rule approximation.

Step 2: Simpson's Rule

Simpson's rule is then used to approximate the value of the integral. The general formula for Simpson's rule is: \[S_n=\frac{h}{3}[f(a)+4\sum_{i=1}^{n/2}f(a+(2i-1)h)+2\sum_{i=1}^{(n/2)-1}f(a+2ih)+f(b)]\]where \(h = (b-a)/n\) is the width of each of the \(n\) subintervals of [a, b]. Given that \(n=4\), \(a=0\), \(b=2\) and \(f(x) = x \sqrt{x^{2}+1}\), substitute these values into the formula to get the Simpson's Rule approximation.

Step 3: Exact Value of the Definite Integral

To find the exact value of the integral, an antiderivative of the function \(x \sqrt{x^{2}+1}\) is needed. Using the power rule for integration, the antiderivative can be found, followed by applying the Fundamental Theorem of Calculus to find the exact value of the integral from 0 to 2.

Step 4: Compare the Approximations and Exact Value

Now compare the value obtained by the Trapezoidal rule, Simpson's rule and the exact value of the integral to understand the approximation errors in the numerical methods.

Key Concepts

Trapezoidal Rule

The Trapezoidal Rule is a numerical method used to estimate the definite integral of a function. It divides the total area under the curve into trapezoids rather than rectangles. This method is useful when an explicit antiderivative is not feasible or when calculations by hand are impracticable.
The formula for the Trapezoidal Rule is: \[T_n=\frac{h}{2} \left[f(a) + 2 \sum_{i=1}^{n-1} f(a+ih)+f(b)\right]\]where \(h = (b-a)/n\) is the width of each subinterval. By applying this to the function \(x \sqrt{x^{2}+1}\), with limits 0 and 2, and \(n=4\), the width \(h\) comes out as 0.5.

By substituting these values into the formula, you can compute the trapezoidal approximation. For functions that curve, this method provides a good estimation; however, the result can still exhibit some error, as the top sides of trapezoids may not align exactly with the curve.

Simpson's Rule

Simpson's Rule is another powerful technique for numerical integration. It provides a better approximation than the Trapezoidal Rule, especially for functions that are reasonably smooth across the interval. Simpson’s Rule uses parabolic arcs instead of straight lines, helping to fit the curve more closely.
The formula for Simpson's Rule is:\[S_n=\frac{h}{3}[f(a)+4\sum_{i=1}^{n/2}f(a+(2i-1)h)+2\sum_{i=1}^{(n/2)-1}f(a+2ih)+f(b)]\]To apply this, consider the interval from 0 to 2 with the function \(x \sqrt{x^{2}+1}\) and \(n=4\), leading to a subinterval width, \(h\), of 0.5.

Simpson’s Rule often yields more accurate approximations by adding weighted contributions of function values at specific points and can reveal insights when balancing cost and accuracy in computations.

Definite Integral

A definite integral calculates the accumulation of quantities, represented physically by the area under a curve for a specific interval. Given by notation \[\int_{a}^{b} f(x) \, dx\]this integral is essential for finding exact values, describing systems in both natural and applied sciences.
The definite integral of the function \(x \sqrt{x^{2}+1}\) from 0 to 2 can be found by evaluating its antiderivative at the boundaries, and applying the Fundamental Theorem of Calculus, which relates derivatives to integrals.

This process can yield a precise value, against which approximations like the Trapezoidal Rule or Simpson's Rule can be compared. This exact value is crucial in understanding how close numerical methods can approach real scenarios and equations.

Approximation Errors

Approximation errors are key elements in numerical integration. They indicate how much an estimated value deviates from the true value. The errors help in understanding and improving the precision of methods such as the Trapezoidal Rule and Simpson's Rule.
These errors occur because numerical methods cannot perfectly align with all curves, especially in sections where the function changes rapidly or has higher curvature.

• The Trapezoidal Rule may overestimate or underestimate significantly, depending on the curve's shape.
• Simpson's Rule tends to yield smaller errors as it accommodates changes in slope more effectively, but this depends on the curve's behavior across intervals.

By calculating and comparing the exact value of an integral and its approximations, one can gauge the accuracy of these methods and determine which is better suited for a given problem.