 Welcome back everyone to our final part of lecture 47 for Math 1210 and lecture one for Math 1220, calculus one versus calculus two. I just wanna mention a very quick limitation about the fundamental theorem calculus. We're gonna come to love the fundamental theorem calculus because it helps us calculate these in, are these definite integrals using anti-drutas and avoids these calculations regarding limits of Riemann sums, which are very, very technical ones right here. Now we're gonna see that anti-drivetas can be difficult to compute as well, but they're a whole lot better than working with limits of Riemann sums. Now, when it comes to using the fundamental theorem of calculus, it's important to remember that there is a limitation. So let me kinda see, let me kinda point out to you in this example, right? If you calculate this integral from negative one to three of the function one over square DX, if you just jump into the calculation without thinking it through, an anti-derivative of one over X, because we could think of this as, we could think of the function one over X squared as X to the negative two. You can use the anti-derivative power rule and you're gonna get X to the negative one over one as you see here. Simplifying that's just the same thing as negative one over X to evaluate from negative one to three. Well, if you plug in three, you're gonna get negative one third. If you plug in negative one, you'll end up with a positive one, but you're subtracting it. So you get negative one third minus one, which is equal to negative four thirds. Now I wanna mention to you that this is not the correct answer. This is total malarkey. Baloney, that is super stinky. Instead, what happens is that the integral from negative one to three of one over X squared DX, this is actually equal to infinity. And to see why that is, it's helpful to actually consider the function. And maybe I should have put a picture in here, but it's okay, I can draw it myself. So we have this function one over X squared. It would look something like the following. One thing to notice here is that this function has a vertical asymptote at the X axis. And so if we go from negative one to three, we are trying to capture this region right here and this region right here. And so as you go towards that vertical asymptote, these rectangles is a taller and taller and taller and the area under the curve gives us infinity. So what's the difference here? Why did the funnel theorem calculate does not work? And the issue has to do with continuity. Is the function continuous on the interval negative one to three? And the answer, of course, for this function is no. It's a big, fat, stinky no. Our function wasn't continuous. And if your function's not continuous, the funnel theorem of calculus does not apply because if you try to integrate discontinuous functions, you could potentially run into problems like this one or things that are far, far more worse. So I don't want you to just assume because the homework typically is giving you only continuous functions that you just run into anti-driven calculations blindly. We have to make sure that the interval in question, this negative one to three, we wanna make sure our functions continuous because if not, we can't use the funnel theorem of calculus and we can get some errors. That actually finishes our discussion about the funnel theorem of calculus. We're gonna use it all the time. So it's not like it's going away, but that finishes our formal lecture about the funnel theorem of calculus here. Thanks for watching. If you like what you've seen, feel free to hit the like button, subscribe if you wanna see more videos like this. And as always, if you have any questions, post them in the comments below. I love to see those questions and I'm glad to answer them whenever you post them. I will see you next time, everyone. Continue to calculate. Bye.