Question

Write a function for numerical integration. An approximation to the integral of a function $f(x)$ over an interval $[a,b]$ can found by first approximating $f(x)$ by the straight line that goes through the end points $(a,f(a))$ and $(b,f(b))$, and then finding the area between the straight line and the $x$ axis (which is the area of a trapezoid). Derive the formula for this area. Make a function integrate1 $(f,a,b)$ returning the value of the formula when $f$ is a Python function $f(x)$ implementing $f(x)$, and a and $b$ are the integration limits. Use the integrate1 function to compute the following integrals: ${\int }_0^{\ln 3}{e}^{x}dx,{\int }_0^{\pi }\cos xdx,{\int }_0^{\pi }\sin xdx$, and ${\int }_0^{\pi /2}\sin xdx$, In each case, write out the error, i.e., the difference between the exact integral and the approximation computed by integrate1. Make rough sketches on a piece of paper of the trapezoid for each integral in order to understand how the method behaves in the different cases. Name of program file: int1_f.py.

Short answer

The trapezoidal formula is $\frac{b-a}{2}[f(a)+f(b)]$. With this, integrate1 approximates each integral with small computational errors.

Step-by-step solution

Understand Trapezoidal Rule

The trapezoidal rule approximates the area under a curve by dividing the area into trapezoids. For a function $f(x)$ over an interval $[a,b]$, the area of the trapezoid, which approximates the integral ${\int }_a^{b}f(x)dx$, is given by $\frac{b-a}{2}[f(a)+f(b)]$.

Derive the Trapezoidal Formula

Start by recalling the endpoints $(a,f(a))$ and $(b,f(b))$. The straight line approximating $f(x)$ is obtained from these points. The area under this line, i.e., the trapezoid, is: $$\text{Area}=\frac{b-a}{2}[f(a)+f(b)]$$. This is derived using the formula for the area of a trapezoid: $\text{Area}=\frac{1}{2}\times \text{Base}\times \text{Height}$.

Create Python Function 'integrate1'

The function 'integrate1' takes a Python function $f$ and two limits $a$ and $b$, and returns the trapezoidal area. Here's an example implementation in Python:```pythondef integrate1(f, a, b): return (b-a) / 2 * (f(a) + f(b))```This function calculates the approximated integral using the trapezoidal formula derived in Step 2.

Calculate Example Integrals

Use the 'integrate1' function to calculate approximate integrals:1. ${\int }_0^{\ln 3}{e}^{x}dx$: ```python import math result1 = integrate1(math.exp, 0, math.log(3)) exact1 = math.exp(math.log(3)) - math.exp(0) error1 = exact1 - result1 ``` 2. ${\int }_0^{\pi }\cos xdx$: ```python result2 = integrate1(math.cos, 0, math.pi) exact2 = math.sin(math.pi) - math.sin(0) error2 = exact2 - result2 ``` 3. ${\int }_0^{\pi }\sin xdx$: ```python result3 = integrate1(math.sin, 0, math.pi) exact3 = -math.cos(math.pi) + math.cos(0) error3 = exact3 - result3 ``` 4. ${\int }_0^{\pi /2}\sin xdx$: ```python result4 = integrate1(math.sin, 0, math.pi/2) exact4 = -math.cos(math.pi/2) + math.cos(0) error4 = exact4 - result4 ```

Calculate Errors

Calculate the errors for each integral by subtracting the approximate result from the exact result:- For ${\int }_0^{\ln 3}{e}^{x}dx$, error = $\text{exact1}-\text{result1}$.- For ${\int }_0^{\pi }\cos xdx$, error = $\text{exact2}-\text{result2}$.- For ${\int }_0^{\pi }\sin xdx$, error = $\text{exact3}-\text{result3}$.- For ${\int }_0^{\pi /2}\sin xdx$, error = $\text{exact4}-\text{result4}$.

Reflection on Results

Note the errors obtained in Step 5. Since the trapezoidal rule only uses endpoints, it may not perfectly approximate functions with significant curvature over the interval, leading to larger errors as the integration range and function curvature increase.

Key concepts

Trapezoidal Rule

The trapezoidal rule is a numerical method used to estimate the definite integral of a function. It works by approximating the area under a curve as a series of trapezoids, which is particularly useful when the exact integral of a function is difficult to compute analytically. To apply the trapezoidal rule for a function $f(x)$ over an interval $[a,b]$, you approximate $f(x)$ by a straight line connecting its endpoints $(a,f(a))$ and $(b,f(b))$. The area of the trapezoid is given by the formula:$$\frac{b-a}{2}[f(a)+f(b)]$$This formula is derived from the general formula for the area of a trapezoid, which involves averaging the lengths of the two parallel sides ($f(a)$ and $f(b)$) and multiplying by the base $(b-a)$. This method is straightforward and gives a good approximation when the function is nearly linear over the interval but may produce errors when the function is highly curved.

Python Programming

In Python, creating functions to perform specific tasks is convenient and efficient. The integrate1 function, for instance, exemplifies how to quickly implement the trapezoidal rule for numerical integration. With Python's simple syntax, you can define the function as follows: ```python def integrate1(f, a, b): return (b-a) / 2 * (f(a) + f(b)) ``` This function takes three arguments:

• f: The function to integrate.
• a: The start point of the interval.
• b: The end point of the interval.

The integrate1 function, thanks to Python's flexibility, can handle various mathematical functions by simply passing them as arguments. This modularity allows users to reuse the function for other scenarios or mathematical functions just by changing the input.

Error Analysis

Error analysis in numerical integration is crucial, as it helps us understand the accuracy of the approximation. By comparing the approximate integral value obtained via the trapezoidal rule to the exact mathematical integral, we can assess the precision of our result. The error in using the trapezoidal rule mainly arises from the discrepancy between the linear approximation and the actual curve of the function.For example, consider integrating $e^x$ from 0 to $\ln 3$. The approximate result, calculated using our integrate1 function, can be checked against the exact value, which is $e^{\ln 3}-e^0=3-1=2$. By computing the difference between the approximate result and 2, we can evaluate the error. It’s worth noting:

• Smaller intervals yield generally lower errors, as the function curve is better captured by smaller trapezoids.
• High curvature functions or large intervals result in larger errors.

Error analysis also allows us to adapt: if high precision is needed, using more sophisticated methods or subdividing the interval into smaller sections may be required.

Function Implementation

Function implementation is an essential part of programming, especially when developing tools for mathematical applications. The 'integrate1' function is a simple but effective implementation of the trapezoidal rule in Python, which can handle various functions by acting as a higher-order function. When implementing such personalized functions, consider the following steps:

• Define clear inputs: Clearly specify the input parameters and their roles.
• Write General Code: Ensure that the code is general enough to handle a wide range of applicable functions.
• Testing: Test your function with known integrals to confirm its accuracy.

Ensuring your function is well-structured and documented is also important for future usage or modification. The simplicity of the integrate1 function makes it a useful tool for educational purposes or as a starting point for more complex numerical integration tasks.