Express the following limit as a definite integral on the interval [0,10][0,10].limn→∞∑k=1n(xk∗−3)2Δx\lim_{n\to\infty}\sum_{k=1}^{n}\left(x_{k}^{\ast}-3\right)^2\Delta x

∫ 0 10 ⁣ ( x − 3 ) 2 d x \int_0^{10}\!\left(x-3\right)^2\,dx

Express the following limit as a definite integral on the interval [ 0 , 10 ] [0,10] . lim n → ∞ ∑ k = 1 n ( x k ∗ − 3 ) 2 Δ x \lim_{n\to\infty}\sum_{k=1}^{n}\left(x_{k}^{\ast}-3\right)^2\Delta x

So in this problem we're asked to express the following limit as a definite integral, and that's gonna be on the interval from zero to 10. So let's go ahead and see how we can solve this. Now what I can see is this is the function we have right here. We have x of k star minus three squared multiplied by delta x, and then we have this limit as n approaches infinity for this summation right here. Now what this in essence is is a Riemann sum that has no end to it. We have unlimited amounts of subintervals, and that's going to give us a truly accurate area under the curve of the function. Now recall that there's another way that we can represent this. What we can do is represent this as an integral of our function right here. Now we don't have to worry about this k stars thing since we're not incrementing up anymore. We're finding a true area. So this whole thing is just gonna be x minus three squared as the function all with respect to x. Now of course, what what's gonna be on the bounds of this integral? Because if we're doing a definite integral, we can't just leave it like this. We have to put some numbers on here. And notice our interval goes from zero to 10. So since that's the place we're finding the area, we're going to go at a low bound of zero to a high bound of 10. And this right here would be the solution. That is how we could represent the following limit as a definite integral given this interval right here. So that is how you can solve these problems where you need to set up a definite integral. Hope you found this video helpful, and let's move on.

Express the following limit as a definite integral on the interval [ 0 , 10 ] [0,10] . lim ⁡ n → ∞ ∑ k = 1 n ( x k ∗ − 3 ) 2 Δ x \lim_{n\to\infty}\sum_{k=1}^{n}\left(x_{k}^{\ast}-3\right)^2\Delta x

∫ 0 10 ⁣ ( x − 3 ) 2 d x \int_0^{10}\!\left(x-3\right)^2\,dx

∫ 10 0 ⁣ ( x − 3 ) 2 d x \int_{10}^0\!\left(x-3\right)^2\,dx

∫ 0 ∞ ⁣ ( x − 3 ) 2 d x \int_0^{\infty}\!\left(x-3\right)^2\,dx

∫ ∞ 0 ⁣ ( x − 3 ) 2 d x \int_{\infty}^0\!\left(x-3\right)^2\,dx

8. Definite Integrals

Introduction to Definite Integrals

Calculus

8. Definite Integrals

Introduction to Definite Integrals

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Multiple Choice

Express the following limit as a definite integral on the interval $open bracket 0 comma 10 close bracket$ .
$\lim_{n \to \infty} \sum_{k = 1}^{n} {(x_{k}^{*} - 3)}^{2} Δ x$

$integral from 10 to 0 comma open paren x minus 3 close paren squared d x$

$integral from 0 to 10 comma open paren x minus 3 close paren squared d x$

$integral from 0 to infinity comma open paren x minus 3 close paren squared d x$

$integral from infinity to 0 comma open paren x minus 3 close paren squared d x$

Verified step by step guidance

Recognize that the limit expression given is a Riemann sum, which is used to approximate the area under a curve. The expression $ \lim_{n\to\infty}\sum_{k=1}^{n}\left(x_{k}^{\ast}-3\right)^2\Delta x $ represents the sum of areas of rectangles under the curve $ (x-3)^2 $.

Identify the interval over which the definite integral is to be calculated. The problem specifies the interval [0, 10]. This means we are looking at the function $ (x-3)^2 $ from $ x = 0 $ to $ x = 10 $.

Understand that $ x_k^* $ represents sample points within each subinterval $ \Delta x $ of the partition of [0, 10]. As $ n \to \infty $, these sample points become dense, and the sum approximates the integral.

Translate the Riemann sum into a definite integral. The expression $ \lim_{n\to\infty}\sum_{k=1}^{n}\left(x_{k}^{\ast}-3\right)^2\Delta x $ becomes $ \int_0^{10}\left(x-3\right)^2\,dx $.

Set up the definite integral $ \int_0^{10}\left(x-3\right)^2\,dx $ to evaluate the area under the curve $ (x-3)^2 $ from $ x = 0 $ to $ x = 10 $. This integral represents the limit of the Riemann sum as $ n \to \infty $.

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