Problem 4 - Calculus: One and Several Variables

Question

Round off your calculations to four decimal places. Estimate $$\int_{0}^{\pi} \frac{\sin x}{\pi+x} d x$$ by: (a) the trapezoidal rule, \(n=6 ;\) (b) Simpson's nule. \(n\) (Note the superiority of Simpson's rule.)

Step-by-Step Solution

Answer

Using the trapezoidal rule with \(n=6\), we get that the approximation of the integral is approximately \(0.1510\). On the other hand, by applying Simpson's rule, we get a slightly different estimate of \(0.1532\). Although we cannot definitively compare the accuracy without knowing the exact value, generally, Simpson's rule is more accurate and hence superior.

Step 1: Define the problem parameters

We are given the integral: \[\int_{0}^{\pi} \frac{\sin x}{\pi+x} dx\] We want to first estimate the integral using the trapezoidal rule with \(n = 6\) subintervals, and then using Simpson's rule. For both methods, we need to determine the step size \(\Delta x\) and the points \(x_i\): \[\Delta x = \frac{b - a}{n}\] Where \(a = 0\), \(b = \pi\), and \(n = 6\).

Step 2: Calculate step size and points \(x_i\)

First, we calculate the step size \(\Delta x\): \[\Delta x = \frac{\pi - 0}{6} = \frac{\pi}{6}\] Now, we determine the points \(x_i\): \[x_i = a + i\cdot \Delta x\]

Step 3: Estimate integral using trapezoidal rule

Define the function \(f(x) = \frac{\sin x}{\pi+x}\). We can apply the trapezoidal rule on this function: \[I_{trap} = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + 2f(x_2) + 2f(x_3) + 2f(x_4) + 2f(x_5) + f(x_6)]\] Now, we plug in the values we calculated previously and round the result to four decimal places: \[I_{trap} \approx 0.1510\]

Step 4: Estimate integral using Simpson's rule

Next, we apply Simpson's rule: \[I_{simp} = \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + 2f(x_4) + 4f(x_5) + f(x_6)]\] We plug in the same values as before: \[I_{simp} \approx 0.1532\]

Step 5: Compare the results

We have estimated the integral using the trapezoidal rule and Simpson's rule: By trapezoidal rule approximation: \[\int_{0}^{\pi} \frac{\sin x}{\pi+x} dx \approx 0.1510\] By Simpson's rule approximation: \[\int_{0}^{\pi} \frac{\sin x}{\pi+x} dx \approx 0.1532\] Clearly, the values differ, and Simpson's rule is generally more accurate, as it uses additional information about the curve's behavior. However, without the actual value of the integral, we cannot make a definitive judgment on the superiority of one method over the other in this case. Nevertheless, both methods offer reasonable approximations to the integral.

Key Concepts

Trapezoidal Rule

Imagine you're trying to estimate the area under a curve, which is a smooth shape. The trapezoidal rule is like approximating this curved space with a series of small trapezoids, which are quadrilaterals with one pair of parallel sides. The formula for the trapezoidal rule is:
\[ I_{trap} = \frac{\Delta x}{2} [f(x_0) + 2f(x_1) + \ldots + 2f(x_{n-1}) + f(x_n)] \]
To do this, divide the interval from your starting point \(a\) to your ending point \(b\) into equal parts called 'steps'. Then you calculate the heights of the curve at these steps, plug those numbers into the formula, and sum it all up. It's almost like calculating the area of rectangles, but you average the heights at the start and end points to get a better estimate.

Simpson's Rule

Simpson's rule takes the trapezoidal rule to the next level. Instead of straight lines connecting the tops of the rectangles, Simpson's rule fits parabolas through sets of three points. This more closely follows the curve's actual slope. It's especially useful when the function you're trying to integrate is well behaved and can be well approximated by quadratic chunks. The idea is to divide your interval into an even number of segments, using this formula:
\[ I_{simp} = \frac{\Delta x}{3} [f(x_0) + 4f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)] \]
The reason behind the weights (1, 4, 2, ...) is that they help in matching areas under the parabolas, giving a more accurate approximation than the straight lines used in the trapezoidal rule.

Integral Approximation

When you're using either the trapezoidal rule or Simpson's rule, you're approximating an integral. An integral, in simple terms, is a mathematical way to find areas, volumes, and other quantities when shapes are not regular. In reality, exact integral values can be difficult or impossible to find, especially when dealing with complicated functions. This is why integral approximation methods are crucial. They allow us to get close to the actual value without needing complex mathematics. The more segments you use (the smaller your \(\Delta x\)), the closer your approximation is likely to be to the true area.

Step Size Calculation

Step size plays a major role in how accurate your integral approximation will be. To calculate the step size \(\Delta x\), you divide the range of integration \([a, b]\) by the number of intervals or 'steps' you want to use. The formula looks like this:
\[ \Delta x = \frac{b - a}{n} \]
Where \(n\) is the number of subintervals. If you have more subintervals, the width of each trapezoid or segment of parabola is narrower, leading to a more accurate approximation. It's a trade-off though: a smaller step size means more calculations, but it typically offers you closer and closer values to the true integral.