The value of lim_{n to infty} left[ frac{1}{n | vedclass

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(D) The given expression is a Riemann sum of the form $\lim_{n 	o \infty} \sum_{r=1}^{n} \frac{1}{n} f\left(\frac{r}{n}ight)$,where $f(x) = \sin x

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$\frac{1}{3}(1 - \cos^3 1)$

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(D) The given expression is a Riemann sum of the form $\lim_{n 	o \infty} \sum_{r=1}^{n} \frac{1}{n} f\left(\frac{r}{n}ight)$,where $f(x) = \sin x \cos^2 x$.This limit is equal to the definite integral $\int_{0}^{1} \sin x \cos^2 x \, dx$.Let $t = \cos x$,then $dt = -\sin x \, dx$,or $\sin x \, dx = -dt$.When $x = 0$,$t = \cos 0 = 1$. When $x = 1$,$t = \cos 1$.Substituting these into the integral:$\int_{1}^{\cos 1} t^2 (-dt) = \int_{\cos 1}^{1} t^2 \, dt$.Evaluating the integral:$\left[ \frac{t^3}{3} ight]_{\cos 1}^{1} = \frac{1}{3} (1^3 - \cos^3 1) = \frac{1}{3}(1 - \cos^3 1)$.

The value of $\lim_{n \to \infty} \left[ \frac{1}{n}\sin \left( \frac{1}{n} \right)\left( \cos \left( \frac{1}{n} \right) \right)^2 + \frac{1}{n}\sin \left( \frac{2}{n} \right)\left( \cos \left( \frac{2}{n} \right) \right)^2 + \dots + \frac{1}{n}(\sin 1)(\cos 1)^2 \right]$ is

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