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Trlangle has vertlces 0)0), 21), and '0)(6 Flnd Its area...

Question

Trlangle has vertlces 0)0), 21), and '0)(6 Flnd Its area

trlangle has vertlces 0) 0), 2 1), and '0) (6 Flnd Its area



Answers

In Exercises $1-6,$ set up the definite integral that gives the area of the region.
$$
\begin{array}{l}{y_{1}=x^{2}-6 x} \\ {y_{2}=0}\end{array}
$$

Of applications of integration. And we're gonna talk about the area between two curbs our first curve use. Yeah, X squared minus six X in our second curve is zero. Okay, we're going to try to find the area between those two. Certainly helps Thio kinda have a vigil for that. So I'm gonna type in X squared minus six X, then mhm zero. It's just gonna be the X axis. Um, show it very well. Why equals zero? Okay. Kind of highlighted in blue. What we need to see here is that the area between these two curves this area right in here is what we're trying to find. We're gonna try to set up an integral to show that on what we What we really need to notice is that the, um, the second equation White will zero is actually on top, and then the parable of the first equation is on bottom. So when we go back here, we set up this integral. We're actually gonna want to integrate. See? We can go back and look at this picture. We can see that this is happening between zero and six. Okay, So from 0 to 6, we're gonna take the top curve, which was zero minus the bottom curve, which waas X squared minus six X. We're gonna integrate all of that from 0 to 6. Clean this up a little bit. We'll get negative X squared plus six x dx dx up here anyway, that's Ah, pretty, pretty basic example of how to find the area between two curbs.

Of applications of integration. And we're gonna talk about the area between two curbs our first curve use. Yeah, X squared minus six X in our second curve is zero. Okay, we're going to try to find the area between those two. Certainly helps Thio kinda have a vigil for that. So I'm gonna type in X squared minus six X, then mhm zero. It's just gonna be the X axis. Um, show it very well. Why equals zero? Okay. Kind of highlighted in blue. What we need to see here is that the area between these two curves this area right in here is what we're trying to find. We're gonna try to set up an integral to show that on what we What we really need to notice is that the, um, the second equation White will zero is actually on top, and then the parable of the first equation is on bottom. So when we go back here, we set up this integral. We're actually gonna want to integrate. See? We can go back and look at this picture. We can see that this is happening between zero and six. Okay, So from 0 to 6, we're gonna take the top curve, which was zero minus the bottom curve, which waas X squared minus six X. We're gonna integrate all of that from 0 to 6. Clean this up a little bit. We'll get negative X squared plus six x dx dx up here anyway, that's Ah, pretty, pretty basic example of how to find the area between two curbs.

Finding the area between two curbs and this one thief equations don't look too intimidating, but it is going to be somewhat complicated because when you can see when I graft these two functions that they do cross three different times, they intersect each other three different times. So we're gonna have to represent this, uh, shaded region with one in a role. We're gonna have to represent this shaded region with another integral but color code. Those it's the first thing we're gonna do is set up an integral for the black region. And what's it is gonna go? Let's establish how far it how far these intersection points are. So the interval of integration here is gonna be from negative 1 to 0, the two x values. And then I'm gonna need the top curve top curve minus the bottom curve. And the top curve is clearly the cubic function excuse minus X. And then the bottom curve is zero. And it's that's it doesn't really affect anything to subtract zero but its throat in their thio to make it very clear and then the red function or the red area that we're trying to represent it. We're going Thio. We're gonna add to this because we're trying to represent the whole thing. So it's the black one with the red one and the red one travels from zero toe one. It's X values. It's interval. And But this time the top curve is now the horizontal line. Um zero minus the cubic function three execute minus sex. You're the zero does mean something because it's gonna change the signs of everything after it. Okay, so let's clean this up. Put these together and simplest terms. So we're going from negative 1 to 0. We'll go ahead and distribute the the quantity of their three X cubed minus three X dia. Great. That one. Yeah, groups of one. It will be. We're just gonna distribute this negative sign with three. So negative three x cubed plus three X That should give us an integral or an integral representation of the area between these two curves. Um, actually, we could Of course, it really doesn't matter if we can. We can leave the the three out in front, also respond to show another their wages that you can be confident it either. The widest correct. Yeah. Mhm. There you go. Either one of those answers air perfectly fine.


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