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3.0252.0151.00.50.20.40.60.8L.00.20.40.60.810Find the area of the region. 0.2...

Question

3.0252.0151.00.50.20.40.60.8L.00.20.40.60.810Find the area of the region. 0.2

3.0 25 2.0 15 1.0 0.5 0.2 0.4 0.6 0.8 L.0 0.2 0.4 0.6 0.8 10 Find the area of the region. 0.2



Answers

Find the area of the shaded region in Exercise $25(\mathrm{d})$

And this problem. We're finding the area on the dartboard that I colored in in blue. It's where our is 3 to 33 and 3/4 toe. Articles four and theta is from nine. Pi over 20. Thio 11 powers 20. Okay, sort of Find the area. You need to do the double in a girl of Juan. Our DRD fada are is going from three and 3/4 which is 15. Force 24 and theta nine pi over 22 11 pi over 20. All right. The integral of our is r squared over two. So now we have nine pi over 20 11 pi over 20 R squared over two from 15. Force toe four. Do you data Mhm. Uh huh. Okay, so we plugged those in. I'm just gonna put the one half out in the front here. Hopes in a girl 16 minus 2. 25/16. Well, d theta. And then you integrate with respect to data and you just get data, um, 16 times 16 uh, 248 off. 248 16 36 412 56 2 56 minus 2. 25. Over 16. Then when you integrate with respect to theta, you get data from 11 pi over 22 99 pi over 20,000 backwards anyway. 11 pi over 20 minus nine pi over 20. Okay, so you end up with one half times 31/16 times two pi over 20 and the twos canceled and you get 31 pi over 320.

In this problem, you're given three points. Told that these points form a triangle and asked to provide the area of that triangle. And so I first step in. This is going to be to find the cross product of the vectors. P Q and P. R. I said. To do this, we need the vectors P Q and P R and to find those. So it's a final elector Peak. You were going to take the point p and subtract or the point Q. I should say and subtract the point p from that s So we're going to take 00 negative one and subtract 210 from that. And the resulting vector is going to be negative to negative one negative one. And then to get the vector PR, we're going to do the same thing. Except we're going to take our and subtract p from that. And so we're going to get negative 4 to 0 and subtract 210 And when we do that, we get negative. 610 Okay? Never want to take the cross product of these two vectors. When we do that, we first read A matrix with its first row is I drink a. It's second row as a components of P. Q. So that's going to be negative to negative one negative one. This third rule was components of PR later. 610 Okay, no the component for I to get that. We're going to cross out the first calm. Take the determines of this two by two matrix. Immediate negative. One time zero minus negative, one times one or zero minus negative. 10 plus one is one, and we always subtracted Jay's component and we cross at the middle call and take the determines of this To bow to matrix, we get negative to time zero minus negative, one times negative. Six. That's going to be zero minus six or negative. Six. The negatives canceled to create a positive. And then, for Kay's component. We're across out the third column. Take the determinants of this two by two matrix. We get negative two times one minus negative, one times negative. Six. That's negative, too. Minus six. That's going to be negative. Eight. And so we found our vector for a cross product, but she's worn a six and they're gonna be, and now we're going to take the length of this factor that will give us the area of a parallelogram which will help us by in the area of the triangle is we're going to take the length of this. And to do that, we're going to take the square room of the son of each component squared. And so we're going to take one squared plus six squared, plus negative eight squared data needs 4 to 64 6 squared is 36 64 plus 36 is 100 plus one square to just one of 100 plus one is 101 and so this is equal to the square roots of 101. So again, this tells us the area of the parallelogram. However, we want the area of a triangle. To do this, we're going to take the area we just found and multiply it times one have. And so our final answer is going to be Route one a one over to and relieving the route one on one as it is because we cannot pull anything out of it. And that is our final answer.

Right. This problem. We need to find area of the shaded regions. So buy formula. We know. Ah, if we want to inter grow instead ofthe X first of the red on the integral. So the area Hey, no e coast too into grow in terms ofthe X And we know the shaded area X goes from zero to one so we can write the bound it pounds from zero to one. Okay. Ah, for here. Ah, the formula is upper curve minus the lower occur. So this will be you two The power X linus x times e to the power X square. So our next step basically is Try toe Aah! Evaluate this, integral. All right, So we can do it. Except to separately the first term. Well, the ranger grow from zero to one into the power ext the ex minus integral from zero to one times to the power X square, the ex. Okay. And the first that we know the anti directly about that. This is e to the Power X. So this is such a yeah to the power X evaluate acts because one minus X equals zero. Okay. And this term here we can bring this acts to the differential that will give us X square. So before we do that, we need two times two. So what we trick they're using here every times two and it divided by two. So the second term will be behalf ties into grove the roads one you two the power X square King X square. And as we can see this if we use change of arable say this explorer to be another variable t Then this is you to the party. So the auntie do it Who will be itself. So this term Ah, you free right here. No B negative One half times into the power X square evaluated and ex ecos one minus x Go to zero. Right? So, in other words, this is, um So next page the first term, remember, is need to the power axe. You're only right from one minus X equals in Cyril zero um, minus one half he to the Power X square. Valerie had taxi. Could one minus X equals zero. So no and serve. It will be the first term. Here is I e to the power of one. Linus Ito, the power zero which is what and the second term and second term is minus one half times the first of eyes that you two, the power one. So it's just e to the tower one minus into a powered zero where she's far and as you can see, answer will be one half times minus one, right.


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