Question

Let S(x)= n, if n is less than x is less than n+1 for every integer n, be the staircase function. Note that S(x) is riddled with discontinuities on suitably large domains. Let M, N, be two arbitrary positives integers subject to 0 is less than M is less than N. Determine a formula in terms of M and N for the integral of S(x)dx from m to n. Solve using improper integrals and sigma notation.

Please help, I am having trouble with this question.

Answer 1

The staircase function S(x) takes on the value of the largest integer less than x. To find the integral of S(x) over an interval [M, N], we can consider the area under the staircase function over each subinterval [n, n + 1] for integers n such that M <= n <= N.

The area under S(x) over the subinterval [n, n + 1] can be represented as a rectangle with height n and width 1. Therefore, the integral of S(x) over the subinterval [n, n + 1] is given by n.

Using sigma notation, we can represent the integral of S(x) over the interval [M, N] as follows:

∫_M^N S(x)dx = ∑_{n=M}^{N-1} ∫_n^{n+1} n dx = ∑_{n=M}^{N-1} n * (n+1 - n) = ∑_{n=M}^{N-1} n = (M + (M + 1) + ... + N-1) = (N^2 - M^2)/2.

So, the integral of the staircase function S(x) over the interval [M, N] is equal to (N^2 - M^2)/2.