Screencast 4.4.2: Fundamental Theorem of Calculus with power functions

Screencast 4.4.2: Fundamental Theorem of Calculus with power functions


[MUSIC] Hello, and welcome to screen cast today about the fundamental theorem of calculus. And this screen cast is gonna be focusing on power functions. Okay, so the fundamental theorem of calculus, as I wrote out here, says that if f is a continuous function on the interval from a to b, and big F is any anti-derivative of f. Okay, cuz remember these come in families, then the integral from a to b of little f of x, dx is equal to big F of b minus big F of a. Okay, so again we’re gonna be focusing here on the idea of the anti-derivative. So that is working our way backwards from the derivative, okay? And we can always check our answers because what you can do is just take the derivative of your answer and make sure you get back to where you started. All right, my first example here today says evaluate the integral from -1 one to 3 of 3x squared minus 2x plus pi. And all of this is gonna be done dx. So remember, that just means then, we are integrating with respect to x, so x is gonna be our variable, okay? All right, we know f is a continuous function, because it’s a polynomial. So we don’t have to worry about anything going funky there. So now we just need to find our anti-derivative. Okay, so let’s take it piece by piece because remember we break this integral up, or we can just do like we did with the derivative whenever we have sums and differences. You can just do the anti-derivative of each piece. Okay, so the anti-derivative of 3x squared. So you’ve got to ask yourself, what function do I take the derivative of that will give me 3x squared? Okay, well the power rule, remember, works in reverse, so that’s gonna be, we need, [COUGH] excuse me, leave our 3 out front, we add 1 this time to our exponent and instead of multiplying that, we’re gonna divide it. Okay, and then hopefully you notice these 3s will cancel, so the anti-derivative of 3x squared is execute, okay. We can check that because the derivative of x cubed Is 3x squared. All right, next piece, so minus the 2 again comes along because it’s a constant, with our constant rule. Then we’d have x, technically this the first power, so if I add 1 I get 2 and I divide by that, I also get 2. So again, this coefficients cancel. So the derivative, sorry, the anti-derivative -2x, is x squared, and that’s because the derivative of x, -x squared, is -2x. All right, and the last one, the constant here plus pi. Well, so now we gotta think to ourselves, all right, so what kind of a function must have a constant slope cuz remember, that was our derivative, of pi? Well, that’s gotta be a linear function, so that’s gonna be the function pi x. And again, if you take the derivative of pi x, do you get back to pi? Absolutely, okay, so let me go ahead and pretty this up a little bit, let me throw my endpoints on here. So we’ve got -1 to 3. So this anti-derivative, then, is x cubed minus x squared, plus pi x, and again I’m going from -1 to 3. Okay, now you notice the dx went away, the integral sign also went away, so these two pieces then are basically the integration idea, okay? But those two pieces do not get carried along, they drop out whenever you do the anti-derivative [COUGH]. Okay, plugging in our numbers then. So we’re gonna have 3 cubed minus 3 squared plus pi times 3. That’s gonna be our f of b basically. So this is the anti-derivative evaluated at our top endpoint minus. Okay, now we gotta definitely gotta watch our negatives on this one. So I’m gonna do -1 cubed minus -1 squared, and then plus pi times -1. So this big quantity here is my big F of a, so that’s my anti-derivative evaluated at my bottom endpoint of -1. Okay, all right so anyway, crunch out some of these numbers, let’s see so this gives us 27 minus 9 plus 3 Pi, okay. Leave that as exact we can, or we can approximate it later, then we have let’s see, -1 cubed will give us -1 minus -1 squared though give us a positive 1, and then that’s gonna be minus pi. Okay, so that gives us, let’s see, 18 plus 3 pi, and then minus -2 minus pi. Combining our like terms, that gives us a negative and a negative, so this negative here has to be distributed, which is why I keep putting these parentheses in here. So that’s gonna end up giving me 20 plus 4 pi. So that’s my exact value of my integral. If you just want an approximate value, though, this gives us about 32.57, if I crunched everything correctly. Okay, fantastic, all right, next example. Again, evaluate the integral. This one looks a little bit uglier, though. So we’ve got the integral from 2 to 5 of 1 over the square root of pdp. Okay, so again, this dp here tells me that I am doing my integral with respect to p. Okay, well if we were doing the derivative of this function I think it was smart to rewrite this first so same thing happens when you do integrals. So we’re gonna have integral from 2 to 5, so those numbers don’t change at all. And remembering the roles from algebra we can rewrite these functions as p to the negative one-half. And then again, that dp gets brought along. Okay, so now this looks more like a power. So I’m going to add 1 to my power, so that gives me negative one-half plus 1, so that’s a positive one-half. And then I’m going to divide by that number, so I’m dividing by a half. And that’s my anti-derivative [COUGH]. Again, I know this because if I do the derivative of this function, I’ll get back to where I started. Okay, I’m evaluating this from 2 to 5, and if we pretty that up a little bit, that’s 2p, actually let’s go and call it 2 square root of p from 2 to 5. So then that’s gonna give us 2 square root of 5 minus 2 square roots of 2. I can’t really do much with those. But if you want that to be an approximate value, that gives us about 1.64, okay? Now, you notice this one had an interval that was very well chosen, okay? Because if you think about this function here, 1 over the square root of p. That function is not continuous. It’s 0 because it’s not defined. But you notice a 0 is not part of my interval on the integral, okay? So that was very well chosen. And just always kind of make sure to double check those things as you’re going, all right? Thank you for watching. [MUSIC]

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