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Problem 18 Easy Difficulty

Express the limit as a definite integral on the given interval.

$ \displaystyle \lim_{n \to \infty} \sum_{i = 1}^n x_i \sqrt{1 + x_i^3} \, \Delta x $, [2, 5]

Answer

$$\int_{2}^{5} x \sqrt{1+x^{3}} d x$$

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Video Transcript

All right, let's go ahead and do this problem. So the question ISS you're given the limit as an approaches to infinity. The some from Aiko's toe want to end off XVIII cleared of one plus x of Ike Uche Delta X okay and X is between two and five inclusive. What is the definite integral form off this expression? 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. So long story short. What's really happening here is that you have a region between a to B, and then you want to calculate the area under the curve using rectangles where this is the height and this is the width. Graphically speaking, this is what's going on. So it just doesn't example if I have a shape that looks like this. This is a this is B. I split it into a bunch of rectangles where this is f of X, Delta X is each of these width in the height is f off X I where the X I is each of these points. Okay, so we know that the area under the curve right here the approximation gets better and better the more rectangles we have. So if we use in rectangles, this is going to be the sum. But as we use the limit as an approaches to infinity, we're gonna have an exact match. So this is the definition off what the definite integral is. Okay, so all we need to do to really answer this question is to see how this is related to that expression right here. It all boils down into figuring out what the limit off the integration is. What effort X is, and de avec dx is usually just Delta X. So we don't need to worry about that part. So here the limit and the some we already know that part is just going to be this s portion of the integral. Now here, X, I is my input. We need to match this as my ex, I What is illegal too? Well, it's X I times the square root of one plus X I Q. Actually, not that difficult. It's exactly what it looks like. This portion becomes your Delta X. So when I multiply f of X, I with Delta X, I get this portion, which is the area under the curve. That's a rectangle. Okay, if we add them all up, it will be the definite integral. And we're taking the limit as and goes to infinity. So we know that it is going to be the exact answer. Now what is the what are the limits off the integration it's given right here? X is supposed to be from 2 to 5, so that is where we get the expression. The integration from 2 to 5 f of x is X square. Root off one plus x you and the exchanges to DX. And this is going to be the answer to this question.