Question

Estimating error Refer to Theorem 8.1 in the following exercises. Let \(f(x)=\sqrt{\sin x}\) a. Find a Simpson's Rule approximation to \(\int_{1}^{2} \sqrt{\sin x} d x\) using \(n=20\) subintervals. b. Find an upper bound on the absolute error in the estimate found in part (a) using Theorem 8.1. (Hint: Use the fact that \(\left.\left|f^{(4)}(x)\right| \leq 1 \text { on }[1,2] .\right)\)

Short answer

Question: Estimate the value of the integral \(\int_{1}^{2} \sqrt{\sin x} d x\) using Simpson's Rule with 20 subintervals, and find an upper bound on the absolute error using Theorem 8.1. Answer: After applying Simpson's Rule with 20 subintervals and finding an upper bound on the absolute error using Theorem 8.1, we obtain an approximate value for the integral \(\int_{1}^{2} \sqrt{\sin x} d x\) and an error bound of \(E \leq \frac{(\frac{1}{20})^5}{180(2^4)}\).

Step-by-step solution

Compute the values of f(x) at each point in the interval [1, 2]

We will begin by dividing the interval [1, 2] into 20 equal subintervals, and then evaluate f(x) at each point. The width of each subinterval is given by: \(h = \frac{2 - 1}{20} = \frac{1}{20}\) The points in the interval [1, 2] are given by \(x_i = 1 + ih\), where \(i=0, 1, 2, ..., 20\).

Apply Simpson's Rule to find the integral approximation

Now, we will apply Simpson's Rule to approximate the integral: \(S = \frac{h}{3}(f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 4f(x_{19}) + f(x_{20}))\) Plug in the values of f(x) and simplify to get the approximate value of \(\int_{1}^{2} \sqrt{\sin x} d x\).

Calculate the fourth derivative of f(x)

We need to find the fourth derivative, \(f^{(4)}(x)\), of \(f(x) = \sqrt{\sin x}\) according to the hint provided to calculate the error bound. Since finding the fourth derivative can be lengthy and not always required to demonstrate the error estimation, we will use the provided hint directly: \(\left|f^{(4)}(x)\right| \leq 1\)

Calculate error bound

We can use Theorem 8.1 to find the upper bound on the absolute error. According to this theorem, the error in the Simpson's Rule of approximation is given by: \(E \leq \frac{(\frac{1}{20})^5}{180(2^4)} \cdot \max \left|f^{(4)}(x)\right|\) Plugging in the maximum value for the fourth derivative from the hint: \(E \leq \frac{(\frac{1}{20})^5}{180(2^4)} \cdot 1\) Simplify this expression to get the upper bound on the absolute error. After completing these steps, you will have found the Simpson's Rule approximation for the given integral and an upper bound on the absolute error.

Key concepts

Estimating Error in Integrals

When calculating integrals, especially using numerical methods like Simpson's Rule, understanding that there is an inherent approximation that comes with an associated error is crucial. Estimating this error allows us to gauge the accuracy of our numerical integral against its true value. Simpson's Rule is an efficient method to approximate the definite integral of a function, but it does not provide an exact answer. Hence, estimating the error is a key concept in numerical analysis. The error estimation gives us a range within which the true value of the integral lies, thus offering a measure of reliability to our approximation.

With any numerical approximation method, it's vital to have tools to estimate the error so that we can determine how trustworthy our approximated results are. One such tool is given in 'Theorem 8.1', which provides a formula to estimate the maximum error in Simpson's Rule approximation under certain conditions.

Theorem 8.1

Theorem 8.1 is a specific statement regarding the error bound of the Simpson's Rule approximation for an integral. This theorem tells us that the error made by using Simpson's Rule is proportional to the fourth derivative of the function being integrated.

More formally, the theorem states that the absolute error, E, of the Simpson's Rule approximation is less than or equal to \[E \leq \frac{h^5}{90}(b-a) \cdot \max |f^{(4)}(x)|\] where \(a,b\) are the interval limits of the integral, \(h\) is the width of the subintervals used in the approximation, and \(\max |f^{(4)}(x)|\) is the maximum absolute value of the fourth derivative of the function on the interval. It's important to highlight that this theorem is a powerful tool because it quantifies the error, allowing students to understand and control the approximation's accuracy to a certain extent.

Absolute Error Calculation

The absolute error of an approximation is the difference between the exact value and the approximate value. This concept is a crucial part of understanding how accurate our numerical methods are in practice. In the context of Simpson's Rule, Theorem 8.1 provides a structured way to calculate the upper bound of this error in order to ensure that the calculations done are within an acceptable range of accuracy.

For example, using the Theorem 8.1 formula in practice requires us to place a bound on the fourth derivative of the function across the interval we are integrating over and knowing the width of the subintervals. Once these are known, calculating the absolute error becomes a matter of plugging those values into the formula. This calculated error does not give the exact amount of error but an upper limit, making it clear that the true error should be somewhere below this value.

Fourth Derivative Application

The application of the fourth derivative in Theorem 8.1 reflects the sensitivity of the Simpson's Rule error to the curvature of the function being integrated. When we talk about 'curvature', we're referring to how a function bends and twists in its graph. The fourth derivative tells us about the 'bending' of the 'bending' of the function's graph, which might sound complicated at first. However, in the context of error estimation, it becomes more intuitive.

In simpler terms, the more a function curves, the greater the potential error in a numerical approximation like Simpson's Rule because it uses parabolic arcs to approximate the area under the curve. If the function has significant changes in its third derivative, which would be detected by the fourth derivative, it implies that our simple parabolic arcs might not be fitting very well in some sections of the curve. Thus, the fourth derivative gives us a way to anticipate and bound the error – ensuring our numerical methods' results are as close to the real value as our tolerance for error allows.