Numerical integration is a method of approximating the definite integral of a function. This is important because there are many functions that cannot be expressed in a closed form, meaning we cannot find an algebraic expression for their integral. In these cases, we must use numerical integration to approximate the integral.

There are two main types of numerical integration: rectangular and trapezoidal.

Rectangular integration is also known as the rectangle rule or left/right Riemann sum. This method approximates the integral by dividing the region under the curve into a series of rectangles and summing the areas of those rectangles. The height of each rectangle is the value of the function at a specific point, and the width is the interval between the points.

For example, consider the function f(x) = x^2. If we want to find the integral of this function from x = 0 to x = 2, we can divide the region into four equal intervals. The function values at these intervals are 0, 1, 4, and 9. We can then calculate the area of each rectangle by multiplying the function value by the interval width, which is 0.5 in this case. The sum of these four rectangles is the approximate value of the integral.

Trapezoidal integration is also known as the trapezoid rule. This method approximates the integral by dividing the region under the curve into a series of trapezoids and summing the areas of those trapezoids. The height of each trapezoid is the average of the function values at the two endpoints, and the width is the interval between the points.

For example, consider the same function f(x) = x^2 from x = 0 to x = 2. If we divide the region into four equal intervals, the function values at these intervals are 0, 1, 4, and 9. The average of the function values at the endpoints of each interval is the height of the trapezoid. The sum of the areas of these four trapezoids is the approximate value of the integral.

Both rectangular and trapezoidal integration are easy to implement, but they have some limitations. For example, they are not very accurate when the function has a lot of curvature. In these cases, more advanced methods such as Simpson’s rule or Gaussian quadrature may be more accurate.

In conclusion, numerical integration is a useful tool for approximating the definite integral of a function when an algebraic expression for the integral is not available. The rectangle rule and the trapezoid rule are two common methods of numerical integration, but more advanced methods may be needed for functions with a lot of curvature.