Пошаговые инструкции
Define the Function and Bounds
First, identify the function \( f(x) \) and the bounds \( a \) and \( b \) of the integral. For example, let's say we want to estimate \( \int_{0}^{2} x^2 dx \).
Choose the Number of Subintervals
Next, choose the number of subintervals \( n \). A larger \( n \) will generally provide a more accurate estimate, but will also increase the amount of computation required. For our example, let's choose \( n = 4 \).
Calculate the Width of Each Subinterval
Calculate the width \( h \) of each subinterval using the formula \( h = rac{b-a}{n} \). For our example, \( h = rac{2-0}{4} = 0.5 \).
Evaluate the Function at Each Point
Evaluate the function \( f(x) \) at each point \( x_i = a + ih \). For our example, we need to calculate \( f(0) \), \( f(0.5) \), \( f(1) \), \( f(1.5) \), and \( f(2) \). Using \( f(x) = x^2 \), we get: \( f(0) = 0 \), \( f(0.5) = 0.25 \), \( f(1) = 1 \), \( f(1.5) = 2.25 \), and \( f(2) = 4 \).
Apply the Trapezoidal Rule Formula
Now, plug the values into the trapezoidal rule formula: \( \int_{0}^{2} x^2 dx \approx rac{0.5}{2} \left[ 0 + 2(0.25) + 2(1) + 2(2.25) + 4 ight] \). Simplifying, we get: \( \int_{0}^{2} x^2 dx \approx rac{0.5}{2} \left[ 0 + 0.5 + 2 + 4.5 + 4 ight] = rac{0.5}{2} \left[ 11 ight] = 2.75 \).
Consider Using a Calculator for Convenience
While the trapezoidal rule can be calculated by hand, it can be tedious and time-consuming, especially for large \( n \). Consider using a trapezoidal rule calculator to quickly and accurately estimate definite integrals. This can be especially useful for checking your work or exploring how the estimate changes as \( n \) increases.
Introduction to the Trapezoidal Rule
The trapezoidal rule is a numerical method used to estimate the value of a definite integral. It works by approximating the area under a curve by dividing it into smaller trapezoids and summing the areas of these trapezoids.
The Trapezoidal Rule Formula
The formula for the trapezoidal rule is: [ \int_{a}^{b} f(x) dx \approx rac{h}{2} \left[ f(x_0) + 2f(x_1) + 2f(x_2) + \cdots + 2f(x_{n-1}) + f(x_n) ight] ] where:
- ( a ) and ( b ) are the bounds of the integral
- ( n ) is the number of subintervals
- ( h = rac{b-a}{n} ) is the width of each subinterval
- ( x_i = a + ih ) are the points at which the function is evaluated
Step-by-Step Calculation
To calculate a definite integral using the trapezoidal rule, follow these steps: