apologiabiology
contestada

show all work and reasoning

which of the following limits is equal to [tex] \int\limits^5_2 {x^2} \, dx [/tex] ?

A. [tex] \lim_{n \to \infty} \sum^n_{k=1}(2+ \frac{k}{n})^2 \frac{1}{n} [/tex]

B. [tex] \lim_{n \to \infty} \sum^n_{k=1}(2+ \frac{k}{n})^2 \frac{3}{n} [/tex]

C. [tex] \lim_{n \to \infty} \sum^n_{k=1}(2+ \frac{3k}{n})^2 \frac{1}{n} [/tex]

D. [tex] \lim_{n \to \infty} \sum^n_{k=1}(2+ \frac{3k}{n})^2 \frac{3}{n} [/tex]

Respuesta :

LammettHash
Split up the interval [2, 5] into [tex]n[/tex] equally spaced subintervals, then consider the value of [tex]f(x)[/tex] at the right endpoint of each subinterval.

The length of the interval is [tex]5-2=3[/tex], so the length of each subinterval would be [tex]\dfrac3n[/tex]. This means the first rectangle's height would be taken to be [tex]x^2[/tex] when [tex]x=2+\dfrac3n[/tex], so that the height is [tex]\left(2+\dfrac3n\right)^2[/tex], and its base would have length [tex]\dfrac{3k}n[/tex]. So the area under [tex]x^2[/tex] over the first subinterval is [tex]\left(2+\dfrac3n\right)^2\dfrac3n[/tex].

Continuing in this fashion, the area under [tex]x^2[/tex] over the [tex]k[/tex]th subinterval is approximated by [tex]\left(2+\dfrac{3k}n\right)^2\dfrac{3k}n[/tex], and so the Riemann approximation to the definite integral is

[tex]\displaystyle\sum_{k=1}^n\left(2+\frac{3k}n\right)^2\frac{3k}n[/tex]

and its value is given exactly by taking [tex]n\to\infty[/tex]. So the answer is D (and the value of the integral is exactly 39).
aliyahyzabelle06

Split up the interval [2, 5] into  equally spaced subintervals, then consider the value of  at the right endpoint of each subinterval.

The length of the interval is , so the length of each subinterval would be . This means the first rectangle's height would be taken to be  when , so that the height is , and its base would have length . So the area under  over the first subinterval is .

Continuing in this fashion, the area under  over the th subinterval is approximated by , and so the Riemann approximation to the definite integral is

and its value is given exactly by taking . So the answer is D (and the value of the integral is exactly 39).

Otras preguntas