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0h) Evaluate the following itegral (Answer Two): (42 ponts) T)y = f dx 2) [= 3) Vzrrn7dxB- Graph and find the area of the region between the X-axis and the curve y...

Question

0h) Evaluate the following itegral (Answer Two): (42 ponts) T)y = f dx 2) [= 3) Vzrrn7dxB- Graph and find the area of the region between the X-axis and the curve y =x -4x where X=3 and X=I: (13 points)

0h) Evaluate the following itegral (Answer Two): (42 ponts) T)y = f dx 2) [= 3) Vzrrn7dx B- Graph and find the area of the region between the X-axis and the curve y =x -4x where X=3 and X=I: (13 points)



Answers

Use the graph of the integrand and areas to evaluate the integral. $$ \int_{-2}^{4}\left(\frac{x}{2}+3\right) d x $$

I have problem for 37 Given r of X, Come on one half x So we integrate x square, do you? Why? Plus two x minus three x y d x that equals the girlfriend 0 to 4 of X squared d d x of one half x plus two x minus three x of one half x The x simplifying This was integral from 0 to 42 x minus X square d x and this comes out to your negative 16/3.

Use the graph to get what this interview is going to be equal to according to theory, is there? So let's go ahead and grab this here. See that the Y intercept for the X. Intercept is up to here. It's mhm. That's all right. And then From here we can see that at 1/2 That's going to equal to three Than at 3/2. It's going to come to one right there so that I have here, if you've got this rectangle right here, So what's on one x 1 are actually a square here. Some of the one x 1 square got he one by two Right triangle. So it's going to be one tips to over two Plus one times 1. So that's going to give us why this one chuckles scenario too.

Okay. What we want to do is we're given in a girl, and the integral is from negative 2 to 4 of X over two plus three d X And what we want to do, first of all, is Graff in a grand. And so that's what we're gonna do if they it's a line. Um and so we're gonna go ahead and Graff that line. Um, and I has a y intercept of three. And it has a slope of 1/2. So down and over, down, over, down and over. And so, um, think that's what she Okay, so that's what it should look like on the line. OK? And now what we want to do is we want to only go from negative too. Two, four. And so we want this area right here. And that area is, um, the area of a trap is oId which is equal to 1/2 be one plus B two times h where B one and B two or your parallel sides. So this is gonna be be one. This will be be too, and this will be a TSH, and so that is going to equal 1/2 be one is a length of two, so that would be too. Plus, be too is a five. And then hide this 123456 long. So this is gonna be equal to 21 square. You'd It's because it's an area.

All right, We're giving this'll problem. It's just all we know, and we have to switch the order of integration. So in order to do that, we need a diagram. It's implied by this immigration. So, uh, we'll start off with Let's take a look at this, Uh, the wise going from negative tune, too. Okay. And the X this changing Cyril to this function. Okay, so we recognize head, That's just a circle in this case since the semicircle. And it is a radius too. It looks like this. So this is X equals square four minus y square. So comes out here to to and the way this is set up, I could drive my strip. Basically, it's this way. No. So it's it's going all way back and forth from Remember, this is why back and forth negative, negative to its and why. And the X goes from zero this way to the curve from actually co Cyril to this curve. So now I want to switch that around. I want to come this way so my ex will be on the outside and I'll be going from they start from zero to to okay left to right zero to maximum. That's his fires. Actual gold from zero to this point. And the why will travel along the strip from the bottom up to the top. So we just need these this equation terms of why So there are two pieces here. Negative four minus X squared and up top we have Why will Serpas it of a four minus X squared under the radical? So those are the two boundaries stand. Why travel from? So we're going to have this, uh, who have why equals negative four minus x square under the ramp and positive for my sex square. And I have a function in here, so that's a switching of the order of integration.


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