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Numerade Educator

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Problem 41 Easy Difficulty

Evaluate $ \displaystyle \int^1_1 \sqrt{1 + x^4}\, dx $.

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Video Transcript

All right, we are going to answer the question. What is the value of one integration from 1 to 1? The square root of one plus X to the fourth the X. Okay, I already know the answer. It only takes half a second to solve this thing. But why can I do that? It's actually based on this idea. The area between the curve and the X axis is written like this. Okay, so one of the first properties that you can see right away is that if the area under the curve it's on Lee from a T A f m x dx, it really doesn't matter what f of X is equal to. So if I try to grow, graph it. Let's say that I have a drawing like this. I started a and I end at a So what's the area under the curve off the line that has, with of zero. Well, I don't know what the height is going to be. I meant I don't know what the function is going to be, but ffx is gonna have ah, height f of A and Delta X is equal to zero. So the area it's just zero because it's f of eight times zero. Okay, So what is this equal to zero? What is this equal to? Well, from 1 to 1, the area under the curve, regardless of what this guy is, has to be zero. And that's how you answer this question.