Numerical Analysis Questions Medium
The trapezoidal rule is a numerical integration method used to approximate the definite integral of a function. It divides the area under the curve into trapezoids and calculates the sum of their areas to estimate the integral.
The basic idea behind the trapezoidal rule is to approximate the curve by a series of straight line segments connecting the points on the curve. These line segments form trapezoids, and the sum of their areas provides an approximation of the integral.
To apply the trapezoidal rule, the interval of integration is divided into equally spaced subintervals. The function values at the endpoints of each subinterval are used to calculate the area of the corresponding trapezoid. The sum of these areas gives an approximation of the integral.
Mathematically, the trapezoidal rule can be expressed as:
∫[a, b] f(x) dx ≈ h/2 * [f(a) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(b)]
where a and b are the limits of integration, h is the width of each subinterval (h = (b-a)/n), and n is the number of subintervals. x1, x2, ..., xn-1 are the equally spaced points within the interval [a, b].
The trapezoidal rule provides a reasonably accurate approximation of the integral for smooth functions. The accuracy improves as the number of subintervals increases. However, it may not be as accurate for functions with sharp changes or oscillations.
Overall, the trapezoidal rule is a simple and widely used method for numerical integration, providing a good balance between accuracy and computational complexity.