Q4Advanced Engineering Mathematics

The objective is to numerically evaluate the definite integral ${\int }_0^1\frac{1}{1+x^2}dx$. While this integral has a simple exact analytical solution ($\arctan (x)$), numerical methods are essential when dealing with non-integrable functions or empirical data. We are tasked with employing both the Trapezoidal Rule and Simpson's 1/3 Rule.

1. Discretizing the Domain

The fundamental first step in any numerical integration scheme is to partition the continuous integration domain $[0,1]$ into a finite number of equally spaced discrete sub-intervals. We will choose a step size (interval width) of $h=0.25$. This creates exactly $n=4$ sub-intervals and $5$ distinct data points. We calculate the corresponding $y$-values ($y=\frac{1}{1+x^2}$) for each $x$-coordinate:

• $x_0=0.00⟹y_0=1.0000$
• $x_1=0.25⟹y_1=0.9412$
• $x_2=0.50⟹y_2=0.8000$
• $x_3=0.75⟹y_3=0.6400$
• $x_4=1.00⟹y_4=0.5000$

2. Applying the Trapezoidal Rule

The Trapezoidal Rule geometrically approximates the area under the curve by connecting adjacent data points with straight lines, forming a series of trapezoids. Its mathematical formulation is $I\approx \frac{h}{2}[(y_0+y_n)+2(y_1+y_2+\cdots +y_{n-1})]$.

3. Applying Simpson's 1/3 Rule

Simpson's 1/3 Rule provides a superior approximation by fitting quadratic parabolic curves through sets of three adjacent points. This rule strictly requires an even number of intervals ($n$ must be even), which is satisfied here ($n=4$). Its formulation is $I\approx \frac{h}{3}[(y_0+y_n)+4(y_{odd})+2(y_{even})]$.

The exact analytical value is $\arctan (1)-\arctan (0)=\frac{\pi }{4}\approx 0.785398$. As theoretically expected due to its higher-order parabolic interpolation, Simpson's 1/3 Rule ($0.7854$) yields a significantly closer approximation to the true exact value than the linear Trapezoidal Rule ($0.7828$).