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Problem 15 Easy Difficulty

Use (a) the Trapezoidal Rule, (b) the Midpoint Rule, and (c) Simpson's Rule to approximate the given integral with the specified value of $ n $. (Round your answers to six decimal places.)

$ \displaystyle \int_0^1 \frac{x^2}{1 + x^4}\ dx $ , $ n = 10 $

Answer

a. $T_{10}=0.243747$
b. $M_{10} \approx 0.243749$
c. $S_{10}=0.243751$

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Video Transcript

So here we have an integral from 0 to 1 of X squared over one plus X to the fourth. We're and equals 10. Um, and our interval with Delta X is going to be equal to B minus a over end. So one minus 0/10, which is your 0.1. So using the trapezoidal rule, what we'll have is Delta X over two. Um, it's gonna look like Delta X over two times F of zero plus two times F of 0.1 plus two times F of 0.2 plus two times F of 0.3 plus two times F of 0.4 plus two times F of 0.5 plus two times f of six You get the picture plus seven eight, all right, and then lastly, we'll have to f of 0.9 plus f of 0.1. When we do all that, what we end up getting Delta X is obviously 0.1. So I've point 1/2. What we end up getting as a result is approximately 0.243747 for our t 10, then for part B we want to do the same method. Only this time it's going to be interval in points. So what that's gonna look like is Delta X times f of X, not plus, um, f of x one. So it's gonna be all those mid points. So that's going to go all the way until f of x nine. So when we do that, it's going to be Delta access 0.1 times. All those values we can calculate and what we end up getting is 0.243748 So he said that these answers are extremely close, which is a great way to estimate integral. Then, for part C, we want to do it using the S method. So what that's gonna look like is S 10 is equal to Delta X over three times F of zero plus four times f of 0.1 plus two times f of 0.2, and this is going to keep going on. It's gonna go 4 to 4 to four to, so that's going to keep going. And then it'll end up with to F of 08 plus four F of 0.9 plus f of one that's going to give us since we know Del Taxes 0.1, we look at our values, we calculate them and we end up getting approximately 0.24 3751 So we see these are all extremely close in approaching the ultimate value that we're looking for.