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Sketch the region enclosed by the curves and find its area.$$y=2+|x-1|, \quad y=-\frac{1}{5} x+7$$

$$24$$

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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Okay. So, um, the absolute value function that they give you is ah, shifted right. One up. Five. Um, and that should make sense because the when you have X minus one an absolute value and then plus five on the outside. Oh, it's well, dyslexia here, I'll shifting up to so scratch what I drew earlier. It's right one up, too, Ariel right here and some v shape can kind of ignore that. That's awful bit, um, and then the other equation is negative. 1/5 X plus seven. Um, so let's say that 67 and that's down one right, five and then up. One left, five to get a curved look. Something like this. Well, a straight line. So if you want to see where these two curves intersect when I said I'm equal to each other. So this right equation is basically the line X plus one, which makes sense if you just, you know, enabling. Plus two you get plus one. If you said that equal to *** 1/5 X plus seven and then say you add 1/5 over. So that would be 6/5 x and subtract one over here six and then multiply by the reciprocal to solve for X, you get five. So that's gonna be my upper bound right here. Uh, at X equals five Vertical line right there. And then you want to do the same thing with this equation, which is negative. X negative. X plus three can see the Y intercept. There's other ways of figuring that out. You want to set that equal to that negative 1/5 X Plus seven Really the same work, but you add 1/5 this way, which make it negative. Four fists Ex Subtract three This way. Seven. Maestri's four and then the same thing multiplied by five negative force times five Negative force. These cancel so X equals negative five. So I have two different colors because I wanted to, um, show the green area. Um, being at X equals negative five. So what I want is the in a role of this, which would be from negative five until one. Because that's where the equation change of the upper function being that negative 1/5 X plus seven minus a lower function, which is the one I wrote earlier that negative X plus three oops, got ahead of myself and ah, when you distribute that negative in there, it's gonna become minus three. So as we combine, if we just simplify that, um, negative 1/5 plus acts will be forfeits. Um, plus four looks very similar to what we had earlier from number 5 to 1 dx weaken. Do the and I deserved of of this, we should be adding one to your exponents. Dividing by two makes it two fists plus four X and that goes from number 5 to 1. Plugging in one into all that you get to fist plus four and then when you plug in negative five into that whole negative five squared is positive. 25 divided by five is just five times two is 10 um, plugging in five or 30 minus 20. So it would get there is probably easiest if I just simplify four minus negative tense. That's 14 and two fists. I'm just gonna leave it like this for right now. So now what I need to dio is this half, which is going from X equals 1 to 5 of a completely different, you know, problem. And then we're gonna add those together. So that's from 1 to 5. The upper functions still the native 1/5 X plus seven. But now, when I subtract off its ah this equation the X plus one equation that we found earlier, some minus X and it's actually minus one because of you have to distribute that minus and so really looks similar to the previous problem. When you combine like terms, Um, from 1 to 5 d x so very similar you add one to your exponents divided by two makes that negative. Three fists, Um, plus six X, and that's from 1 to 5 as a plug in five. And for all that, five squared is 25 divided by +55 times negative. Three is negative. 15 plus six times five is 30 and then minus plugging in one. I feel like I've over explain this problem, but stick with me. I am almost there. Um, the best way to do this, though, is to, you know, you get 15. There it becomes minus six. So that's nine and three fists. And as I had a plus sign up there somewhere, as I add these together to fist plus three fists is one plus nine is 10 plus 14 is 24 way too much work, but that's correct.

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