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. $$\lim _{n \rightarrow \infty} \sum_{k=1}^{n}\left(\sin ^{3} c_{k}\right) \Delta x, \quad[-\pi, \pi]$$

Short answer

The given limit of the Riemann sum represents the definite integral: \(\int_{-\pi}^{\pi} \sin^{3}(x) dx \).

Step-by-step solution

Identify the function and interval in the sum

The function here is \(\sin^{3}(x)\) and the interval is \([-π,π]\). The \(\Delta x\) in the sum represents the difference in x-values between successive subintervals, which is equal to \((b-a)/n\) in a regular partition of the interval [a,b], here \((π-(-π))/n=2π/n\).

Write as Riemann Sum

We can write the sum as follows: \(\sum_{k=1}^{n}\sin^{3}(c_{k}) * \Delta x = \sum_{k=1}^{n}f(c_{k}) * \Delta x\) where \(f(x) = \sin^{3}(x)\) and \( \Delta x = 2π/n\). This is a Riemann sum of the function \(f(x)\) over the interval [-π,π]. The variable \(c_{k}\) represents any sample points chosen in each subinterval.

Express the limit as a definite integral

We know that the definite integral over some interval [a,b] of a function can be represented as the limit of the Riemann sums over that interval as the number of subintervals approaches infinity. Thus the limit can be expressed as the integral: \( \lim_{n \rightarrow \infty} \sum_{k=1}^{n}\left(\sin^{3} c_{k}\right)\Delta x = \int_{-\pi}^{\pi} \sin^{3}(x) dx \).