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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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

Vedclass · Accepted Answer

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

Vedclass · Answer

(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

Similar Questions

Evaluate the following definite integral as a limit of sums: $\int_{a}^{b} x \, dx$

$\lim _{n}$ ${\rightarrow \infty}\left(\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2}+\frac{n}{n^2+3^2}+\ldots+\frac{n}{n^2+(2n)^2}\right)=$

$\lim _{n \rightarrow \infty}\left(\frac{n^{2}}{\left(n^{2}+1\right)(n+1)}+\frac{n^{2}}{\left(n^{2}+4\right)(n+2)}+\frac{n^{2}}{\left(n^{2}+9\right)(n+3)}+\ldots+\frac{n^{2}}{\left(n^{2}+n^{2}\right)(n+n)}\right)$ is equal to

Evaluate the following definite integral as the limit of a sum:
$\int_{0}^{4} (x + e^{2x}) \, dx$

For $a \in \mathbb{R}$ (the set of all real numbers),$a \neq -1$,if $\lim_{n \to \infty} \frac{1^a + 2^a + \dots + n^a}{(n+1)^{a-1}[(na+1) + (na+2) + \dots + (na+n)]} = \frac{1}{60}$,then $a$ is equal to:

Vedclass Test Series

Exam Paper Generator

Online Exam Module