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Sketch the region enclosed by the curves and find its area.$$y=x, y=4 x, y=-x+2$$

$$\frac{3}{5}$$

Calculus 1 / AB

Calculus 2 / BC

Chapter 6

APPLICATIONS OF THE DEFINITE INTEGRAL IN GEOMETRY, SCIENCE, AND ENGINEERING

Section 1

Area Between Two Curves

Integrals

Integration

Applications of Integration

Area Between Curves

Volume

Arc Length and Surface Area

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I think what makes this problem somewhat difficult is all the different, I guess. Visuals. So you have like, was X um, like ALS for X, which is a steep graph. I'm graphing isn't difficult. And then y equals negative X minus two, if you autograph right there. You see Michael's negative Explicit to eso, what's important is finding these X values, which already found earlier. You said the equations equal to each other. Um and so I like fractions. So the X values air 2/5. I wear mine green and blue intersect, and you can double check by plugging in if you plug into fists and for X here you get a fist and, um, to minus 2/5 is a FIS um, and then this X values X equals one, and you can double check that the red and blue intersect right there. Um, X equals one negative. One plus two equals one. And the last one that's obvious is this X equals zero that would been write it anyway, So you're gonna have to different in a girl's the animal from 0 to 2 fists of I think it's kind of obvious I'm doing this area by the way of the green function minus the red function. Well, that's four x minus X. So I hope you're OK. I'm just gonna write three X on that and then the inner rule from two fists to one of the blue function minus the red function. So the blue function being negative X plus two minus x So it's gonna become I hope you're OK. I'm gonna write negative two x plus to his native x minus X is negative two x dx. So now it's just a matter of doing the anti derivative. Well, this one is becomes X squared. And then you have to divide by your new um, exponents. And that's from 0 to 2 fists. And then plus, um well, it's gonna become negative X squared when you divide, like to cancel plus two X, and that is from two fists toe one for that function. So this one's pretty straightforward cause plugging into fists when you square. Each of those two squared is four five squared is 25. Um, the three and the two are still there. But what actually happens here is two goes into four twice. So you're going to see me, right? This is six 25th. Um, and we saw this other one to dio plugging in one's really nice because, uh, it just becomes negative one, uh, plus two. And then you have to subtract off plugging into fists and same as before, um, becomes 4/25. Um, plus 4/5. Um, so as I'm simplifying this, I would actually just write this as name one plus two is one. So that's 25. 25th, because I like getting my denominator to be the same. Same thing with this one. I would probably change this to B times in by five. Five times five is 25 4 times five is 20 on a four plus 20th 16. So I have a minus 16. 25th. So then, from there might be best to just say, Oh, well, six minus 16 is negative. 10 25 minus 10 is ah, 15 25th. I don't know. I keep changing colors. No rhyme or reason why I'm doing that. Simplifies. Did three fists. The answer I would commit to

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