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Problem 3 Easy Difficulty

Find the area under the curve $ y = \frac{1}{x^3} $ from $ x = 1 $ to $ x = t $ and evaluate it for $ t = 10, 100 $, and $ 1000 $. Then find the total area under this curve for $ x \ge 1 $.

Answer

$\lim _{t \rightarrow \infty} A(t)=\lim _{t \rightarrow \infty}\left[\frac{1}{2}-1 /\left(2 t^{2}\right)\right]=\frac{1}{2}$

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Video Transcript

Hello. Welcome to this. Last in this last May. Look at the area under the cab. Why it is called to one uber X, The bath three. Yeah, X go to one, then X costed to because to t Okay, So the area Yeah, is the integral Why'd yeah d x? Okay, it's from one t. So here. Yeah, we have the integral than one over x to the Bathory B X So that is equal to Yeah. Oh, the eggs. So a TV comes No. Okay, so when you're integrating it, you are want with one to the power from one to t. Yeah, so that is excellent. By negative two on negative too. From one to t so we can bring out the negative too. Then we have X to the power. Yeah. Cool. One of, uh, X squared from one to t. So here. Yes, the negative half out. That one over. She squared minus 1/1. So here we have the 80 again. The area is half one over T squared, minus one. Okay, so now that we know it for expression in terms of t, we find this when t is called 100. Then we find it? Us? Yeah. Yeah. So that is, uh, sorry. The that t is standing in the first place, so that is negative. Half one, over 100 minus one. Okay, Okay, so this gives us negative half, then there point there. One minus one. Okay, so the person becomes they opened. Negatives opened. Five, then. Mm. Yeah. Okay, so let is there a coin? 459 You're also looking at it at X, because to mhm AT T, it costs 200 the second time. So that is negative. Half one of the 100 squared, minus one. So that is negative. Right? So 100 squared and this Mhm. Yeah, yeah, yeah. So that is mhm. So the whole Chinese side is negative. Their coin? Yeah. 9994 times. Okay. Mhm. When you multiply that by 0.5. Okay, you have zero point 499 five approaching 0.5. The last but not least, um, the value stereo place is 1000. Okay. Mhm. Yeah. So that is so let's make that negative 0.5. Then. Let's look at the whole thing in there. Mhm. Yeah, yeah, yeah. Mhm. Uh huh. Yeah. So negative. 0.9. Yeah. Okay, six times. Okay, then the whole thing. I'm 0.5. So that's the number of pieces then that approaches. Mhm. Yeah, you have a Okay, so we'll be nice to find there. Area us. This is greater than or equal to one. So here we have. Okay. The limit of the limited t approaching an infinity. Yeah, of eight. Okay, so here we have half out. Yeah. Mm. Yeah. So let's introduce the limit here as well. Yeah. Mhm. Mhm. Okay, So if t squared approach is a very high number, then because it's a denominator becomes zero. Uh, it becomes there with the whole thing. Here. Becomes there. Okay, Cool. What's one of every huge number becomes zero. So this is, um, zero minus one. Okay. And that is half minus times minus one. And that gives us how. Okay, open five. So the limit as t approaches a very huge number, the area becomes half. And as you can see, as you increase t, the area becomes closer to half. Okay, so this is what time would afford us. This is the end of the lesson. Thanks. Your time