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calculus - Express this limit as a definite integral. No interval given. $\lim\limits_{n\to\infty}\sum_{k=1}^n \left(1+\frac{2k}{n}\right)\cdot \frac{2}{n}$

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calculus – Express this limit as a definite integral. No interval given. $\lim\limits_{n\to\infty}\sum_{k=1}^n \left(1+\frac{2k}{n}\right)\cdot \frac{2}{n}$

I would like to correct the answer above/below me, with my own comments.

If the Riemann integral $\int_a^b f(x)\,dx$ exists, then it can be written as the limit of a special sum known as a Riemann sum

$$\int_a^b f(x)\,dx=\lim_{n\to \infty}\sum_{k=1}^n f(c_k) \Delta x \tag 1 $$
where $c_k = a + \frac{b-a}{n} \cdot k $ and $ \Delta x = \frac{b-a}{n}$. The formula for $c_k$ are right endpoints of each of the n uniform width subintervals.

Note that the choice of $c_k$ is not unique and different $c_k$ will produce different functions with different limits for the integral. However the final value for the definite integral should end up being the same.

I will choose $c_k= 0 + \frac{1-0}{n} \cdot k = \frac{k}{n} $ which forces $ \Delta x = \frac{1-0}{n} = \frac 1 n $.

It may seem more natural to choose $c_k= 1 + \frac{3-1}{n} ~ k$ and $ \Delta x = \frac 2 n $ . This will lead to the first answer posted above. I will leave it to you to read that answer.

Using some algebra we can rewrite the original Riemann sum in the appropriate ‘integral ready’ form using our choice $c_k= \frac{k}{n}$ and $ \Delta x =\frac 1 n $:

$$\begin{align}\lim_{n\rightarrow \infty}\sum_{k=1}^n \left(1+\frac{2k}{n}\right)\cdot \frac{2}{n} &= \lim_{n\rightarrow \infty}\sum_{k=1}^n 2\left(1+ 2 \cdot \frac{k}{n}\right)\cdot \frac{1}{n}
\ &=\lim_{n\rightarrow \infty}\sum_{k=1}^n 2\left(1+ 2 \cdot c_k\right)\cdot \Delta x
\ &= \lim_{n\rightarrow \infty}\sum_{k=1}^n f(c_k) \cdot \Delta x
\ &= \int_{0}^{1} f(x) ~dx \end{align} $$

Notice how the $c_k$ becomes the $x$ in the definite integral.
It follows the Riemann Sum is equal to the integral $\int_0^1 2(1+2x)\,dx$.

SOLVED:Express the limit as a definite integral o…

SOLVED:Express the limit as a definite integral o… – So the definite integral, the integral from a to B of ffx the the X this is defined to be the limit of a remonstrance. Some limit as angles to infinity off the some off I equals toe Want to end off f off X I using the same notation Delta X, where Delta X is equal to B minus a over and where the X is defined between. We need to be okay.On a definite integral, above and below the summation symbol are the boundaries of the interval, [a, b]. The numbers a and b are x -values and are called the limits of integration; specifically, a is the lower limit and b is the upper limit. To clarify, we are using the word limit in two different ways in the context of the definite integral.Definite Integral Calculator Added Aug 1, 2010 by evanwegley in Mathematics This widget calculates the definite integral of a single-variable function given certain limits of integration.

5.2 The Definite Integral – Calculus Volume 1 | OpenStax – The integral is "the sum expressed with the sigma symbol as "n" is allowed go to infinity". The "sigma" is then changed to the "music note". A "definite" integral is an integral that can be evaluated between two specific values of a variable. "Definate integral" does not mean writing the summation as an integral as n is allowed go to infinity.so we've got a Riemann sum we're going to take the limit as n approaches infinity and the goal of this video is to see if we can rewrite this as a definite integral I encourage you to pause the video and see if you can work through it on your own so let's remind ourselves how a definite integral can relate to a Riemann sum so if I have the definite integral from A to B of f of X f of X DX weFree definite integral calculator – solve definite integrals with all the steps. Type in any integral to get the solution, free steps and graph This website uses cookies to ensure you get the best experience.

5.2 The Definite Integral - Calculus Volume 1 | OpenStax

Wolfram|Alpha Widgets: "Definite Integral Calculator – Note that the choice of c k is not unique and different c k will produce different functions with different limits for the integral. However the final value for the definite integral should end up being the same. I will choose c k = 0 + 1 − 0 n ⋅ k = k n which forces Δ x = 1 − 0 n = 1 n.Write the limit as a definite integral on the interval [a, b], where ci is any point in the ith subinterval. Express the integral as an iterated integral in six different ways, where E is the solid bounded by the given surfaces. asked Feb 18, 2015 in CALCULUS by anonymous.Express the limit as a definite integral. 73. lim n → ∞ ∑ i = 1 n i 4 n 5 [ H i n t : consider f (x) = x 4.

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