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University of North Texas

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Problem 57 Easy Difficulty

What is wrong with the equation?

$ \displaystyle \int^{\pi}_{\pi/3} \sec \theta \tan \theta \, d\theta = \sec \theta \Bigg]^{\pi}_{\pi/3} = -3 $

Answer

$\int_{\pi / 3}^{\pi} \sec \theta \tan \theta d \theta$ does not exist.

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Video Transcript

So if we want to figure out what is wrong with this equation, Um, first, let's really let's look at what? Seeking Seita Tangent status graph. Looks like on pie 32 pi. So I went ahead and graft this, um, desk most, actually. Maybe let me go ahead and put the interval. So pi third x two. Okay, this is what that looks like. Well, let's zoom out. Well, you can see how all of this is going to be positive, but somehow we get a negative answer. And remember, a negative area means we're below the X access. Um, ever integrating or we have more area underneath the X axis than above. But over here knows how all of this is positive. And we actually have a vertical acid tote at, um on our azure, maybe like pious or something. Um, so that is actually what's going to end up being wrong? One. We know that this should be strictly greater than zero Andi. Even more than that, if we kind of look at this so seeking is supposed to be one of our coastlines. One of our coastline data and then tangent is supposed to be signed over Coast Side. Yeah, so at Seda is equal to pi half, um, seek and feta tangent. Feta is undefined, and it just so happens it looks like it's going to positive infinity. And so because of that, this limit or this integral here should just not exist it all. So this is kind of why it's important to Even though you could integrate this, you have to make sure you don't have anywhere where, like, you can plug in, like, zero or just have any kind of undefined values. So, yeah, this is why, um, this isn't actually going to end up being true.