Step 1 :The given limit is a Riemann sum for the definite integral of the function \(f(x) = \frac{\pi}{4} \tan(x)\) from 0 to \(\frac{\pi}{4}\). This is because the limit of the sum of \(f(x_i)\Delta x\) as \(n \rightarrow \infty\) is the definition of the definite integral of \(f(x)\) from \(a\) to \(b\), where \(x_i\) are points in the partition of the interval \([a, b]\) and \(\Delta x = \frac{b - a}{n}\). In this case, \(a = 0\), \(b = \frac{\pi}{4}\), \(f(x_i) = \frac{\pi}{4} \tan(x_i)\), and \(\Delta x = \frac{\pi}{4n}\). Therefore, the limit is equal to the area under the graph of \(y = \frac{\pi}{4} \tan(x)\) on the interval \([0, \pi / 4]\).
Step 2 :The area under the graph of \(y = \frac{\pi}{4} \tan(x)\) on the interval \([0, \pi / 4]\) can be calculated as \(-\frac{\pi}{4} \log\left(\frac{\sqrt{2}}{2}\right)\).
Step 3 :Comparing the options, we find that the area corresponds to the region under the graph of \(y=\tan (x)\) on the interval \([0, \pi / 4]\).
Step 4 :Final Answer: \(\boxed{\text{D. The area of the region under the graph of } y=\tan (x) \text{ on the interval } [0, \pi / 4]}\)