Riemann Sum in SageMath

Riemann Sum is a method of approximating the area under a curve by dividing the area into rectangles and summing up their areas. SageMath provides a simple way to implement Riemann Sum for a given function.

Implementing Riemann Sum in SageMath

To implement Riemann Sum in SageMath, we need to define a function, specify the bounds of integration, and the number of rectangles we want to use.

f(x) = x^2
a = 0
b = 1
n = 5

In the above code snippet, we have defined the function f(x) = x^2. We want to find the area under this curve between x = 0 and x = 1 using n = 5 rectangles.

Next, we can define a function to calculate the area using Riemann Sum.

def riemann_sum(f, a, b, n):
    dx = (b - a) / n
    x = a
    area = 0
    for i in range(n):
        area += f(x) * dx
        x += dx
    return area

In the above code snippet, we have defined a function riemann_sum that takes in the function f, the bounds of integration a and b, and the number of rectangles n. We calculate the width of each rectangle dx as (b - a) / n. We start at x = a and iteratively calculate the area under each rectangle.

We can now use this function to calculate the area under the curve of f(x) = x^2 between x = 0 and x = 1 using n = 5 rectangles.

area = riemann_sum(f, a, b, n)

The output of this code will be the area under the curve, which is 0.34.

Conclusion

In this tutorial, we have seen how to implement Riemann Sum in SageMath to calculate the area under a curve. This method is useful for approximating integrals where the exact solution is difficult or impossible to calculate. SageMath provides a convenient way to work with mathematical functions and implement numerical methods like Riemann Sum.