Question

Consider the definite integral $D = \int_0^\pi \sin(5x) \, dx$. Which of the following expressions represents the integral $D$ as a limit of Riemann sums? (There may be multiple correct answers; choose all of them to receive full credit). $\boxed{\quad} \lim_{n \to \infty} \sum_{i=0}^{n-1} \frac{\pi}{n} \sin\left(\frac{5\pi i}{n}\right)$ $\boxed{\quad} \lim_{n \to \infty} \sum_{i=1}^n \sin\left(\pi + \frac{5\pi i}{n}\right)$ $\boxed{\quad} \lim_{n \to \infty} \sum_{i=1}^n \frac{\pi}{n} \sin\left(-\frac{5\pi i}{n}\right)$ $\boxed{\quad} \lim_{n \to \infty} \sum_{i=1}^n \frac{\pi}{n} \sin\left(\frac{5\pi i}{n}\right)$

          Consider the definite integral
$D = \int_0^\pi \sin(5x) \, dx$.
Which of the following expressions represents the integral $D$ as a limit of Riemann sums? (There may be multiple correct answers; choose all of them to receive full credit).
$\boxed{\quad} \lim_{n \to \infty} \sum_{i=0}^{n-1} \frac{\pi}{n} \sin\left(\frac{5\pi i}{n}\right)$
$\boxed{\quad} \lim_{n \to \infty} \sum_{i=1}^n \sin\left(\pi + \frac{5\pi i}{n}\right)$
$\boxed{\quad} \lim_{n \to \infty} \sum_{i=1}^n \frac{\pi}{n} \sin\left(-\frac{5\pi i}{n}\right)$
$\boxed{\quad} \lim_{n \to \infty} \sum_{i=1}^n \frac{\pi}{n} \sin\left(\frac{5\pi i}{n}\right)$

Consider the definite integral
D = ∫0^πsin(5x) dx.
Which of the following expressions represents the integral D as a limit of Riemann sums? (There may be multiple correct answers; choose all of them to receive full credit).
limn →∞∑i=0^n-1(π)/(n)sin((5π i)/(n))
limn →∞∑i=1^n sin(π + (5π i)/(n))
limn →∞∑i=1^n (π)/(n)sin(-(5π i)/(n))
limn →∞∑i=1^n (π)/(n)sin((5π i)/(n))

Added by Alyssa R.

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