Question

In Exercises \(1-6,\) each \(c_{k}\) is chosen from the \(k\) th subinterval of a regular partition of the indicated interval into \(n\) subintervals of length \(\Delta x .\) Express the limit as a definite integral. $$\lim _{n \rightarrow \infty} \sum_{k=1}^{n} c_{k}^{2} \Delta x,[0,2]$$

Short answer

\(\int_{0}^{2} x^{2} dx\)

Step-by-step solution

Understand the partition

In the problem, it's mentioned that each \(c_k\) is chosen from the kth sub-interval of a regular partition of the interval [0,2] into n subintervals of length \(\Delta x\). Hence for kth subinterval, \(c_k = \frac{k}{n} * 2\). Here, k can be any integer from 1 to n.

Recognize the Limit of Sum as Definite Integral

The expression \(\lim _{n \rightarrow \infty} \sum_{k=1}^{n} c_{k}^{2} \Delta x\) can be recognized as a Riemann sum as \(n\) approaches infinity, which indicates a definite integral. Therefore, each term in the sum can be considered as an area of a rectangle with height \(c_k^2 = (\frac{k}{n} * 2)^2\) and width \(\Delta x = \frac{2}{n}\).

Express as Definite Integral

As the limit of this Riemann sum as \(n\) approaches infinity is a definite integral, this can be expressed as the integral of the function \(f(x) = x^2\) over the interval [0, 2]. Hence the expression \(\lim _{n \rightarrow \infty} \sum_{k=1}^{n} c_{k}^{2} \Delta x\) on [0,2] equals to \(\int_{0}^{2} x^{2} dx\)