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Sketch the region enclosed by the given curves and find its area.

$ y = \sqrt[3]{2x} $ , $ y = \frac{1}{8}x^2 $ , $ 0 \le x \le 6 $

$\frac{47}{3}-\frac{9}{2} \cdot \sqrt[3]{12}$

Applications of Integration

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Campbell University

Oregon State University

Harvey Mudd College

Baylor University

all right. Sketch the region close by those two curves. Um, in this domain ex belongs Cyril and six. But I say this is our ex biplane and our domains zero six. The first function is Ah, the first function is actually white. If we rewrite this disfunction issue B A to X to the power run third. Okay. No, if made joy, you know the X equals zero vicryl zero, but x equal to six. This eco's tio Ah, twelve to the one third. Right. So this is actually a code six. And this car service is, uh, Kong Cave Or call Max. It's a complex upwards. Right? Because if we take Cube on both sides, this will give us two X equal to y cube. And we know actually close to my cube something like this. Right in the second. The second curve here is of wiii coast, while eight packs square is the problem. So we're actually coaches of X equals zero. This goes to zero. So the first intersection of those two points this region X equals six. This is Ah, thirty six over eight. Right before. So it will be night. Were, um Now, Now we're too right. So Ni over too. And we know X equals six. Why, coz to now we're too, which is four point five for fine. Five is greater than this health to the one third Because we know the cube off to four point five is more than twelve, right? Something like here of Les This will be over Nice over to you or a curve. They'Ll look like this is a problem Here is sorry enclosed region. Okay, you know, order to find this point. Ah, Now let's try to find that the other intersection point X x two so x two shoes satisfy both car. In other words, they need to solve this equation. Teo ex ico c o o r. Eight x square And I mean you get a read off this thing. So we take you Cuba three takes three power on both side is so the left side will give us two acts and the rest side is just Ah one nowhere. Aid to the power three time's act square to parse wave, which is X six and I mean no Act two is now zero, so we can cancel one x So this will return. Acts are fine. Eco's Tio go um ico. So what I see too times eight Cute, right? And they know ate you co two two to the power of three. So this is two to the power of ten. Then you can sell for X. You know x two Must Eco's tio to go into the power too Meesh is for Okay. So, Anna, we see that if we want to calculate the area issue, take the integral with respect to X. Since everything is represented backs, you know, provide ais much more information. Say they take the integral with respect acts. I mean, our exit goes from zero to x two and accessories for things I the integral. Just a precursor minus the lower curved upper curve. Here is is a route to acts and minus o r nor occur our lower curve here ease while we're eight X square. Okay, Now let's find Auntie dear to for this. You know, the auntie do it is that this will be on my third eye formula. Think this beer to x for over three times three or four time's too, right? This will be the Aunt Edie derivative offer a route to axe. Why's that? Because if we take the purity about this thing, we bring down for for one third of four over three up front. So this canceled. The coefficient will be one and on and thiss power here He's forward. Three minus one. It will be a third. And by channel they also need two times two. So, actually, you know what? Here you should be too. Is the wide by two if times well half okay. And second term here is just one or twenty four. Three times eight times. Execute right. This is Auntie dear. Give off our aides X square and every value at the boundary for zero. You know, the actual code zero Both Tommy called zero. So we only care of X equal to four actually could sue for rape. Raining. This is eight, which is to the third to the third. So three canceled. It will be too through the forest. And to know fourth day sixteen or four is for times two and four. Cancel with too. So we get to hear the first army six and second term is sixty four for a Q B sixty for over twenty four. Oh, six or over twenty four. You know, we divide by four developed by eight This give us eight, then give us three. So the answer will be sixteen, ten hour, three.