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Find the area of the given triangular region using vectors.(0.0. 4(0.-4.0(2,0,0)...

Question

Find the area of the given triangular region using vectors.(0.0. 4(0.-4.0(2,0,0)

Find the area of the given triangular region using vectors. (0.0. 4 (0.-4.0 (2,0,0)



Answers

Use integration to find the area of the triangular region having the given vertices. $$ (0,0),(4,0),(4,4) $$

For this problem we are asked to use integration to find the area of the triangular region having the vertex is 0040 and 64 So let's set up some axes here. So we have our 00 point at the cross there. So we have 00. Then we have 40 which would be 1234 right here. And we have 64 We go up to um 56 there and 1234 there. So are triangular region. It's like this. Now the way that we can, the way that we can treat this is splitting this up into two sections. So you can see that we have a small right angle triangle ending at 40 The previous problem indicated that the area of a right hand right angle triangle with the given vertex. Is there actually no? Excuse me? I need to take that back. It's not going to be the same. Never mind. All right. So, ah having the points that we have here, we can see that the slope of the line is going to be will rise of four run of six. So we can reduce that down to M equals to over three. And we can see that clearly the Y intercept is zero. So we have that Y equals to over three X. And we can see that essentially we have two different regions. Area one where the lower bound is just the horizontal axis and area to where the lower bound is going to be. Essentially this extra line which has a rise of four and a run of two. So for that one the slope is going to be too. So, we are looking in area two for the region in between Y equals to over three X and y equals two X. Um And why intercept? I'm just going to do this sort of graphically here. All right. So that would be 123456782 x -8. All right. So having that it should have been obvious with why intercept was not sure why it took me that long. Okay. So having that the area is going to equal the integral from zero up to four of just 2/3 X. D X. Plus. The integral from four up to six of 2/3 X minus two X minus negative eight. So plus eight the X. So first of all integrating to over three X. That's going to turn into 2/3 times X squared over two. Or just X squared over three, evaluated from 0 to 4. That would turn into four squared over three. That's going to be 16/3 for that first area. Plus we'll have another 16/3. Or actually no, I take that back we would have to over three times X squared over two, so we'll have X squared over three minus two times X squared over two, so minus X squared plus eight X, Evaluated from four up to six. So all that's left there is to plug in things and calculate. So give give me one second here. I'll pause the recording, Adding everything up. The total should be eight.

Triangle. With these three points, I'm a plot them really quickly. Here's zero comma. Negative nine here. Zero comma, Negative four. And here's five combination of force. But we have a right triangle right here. Then we want to know what the area is. Area is equal to base times height over two. So we need to figure out what this side is equal to in this side. From this point to this point, this one right here is zero common negative four. And this one is five common negative four. The distance between the X values is five. So this site is equal to five from this point right here to the top 0.0.0 common nine. The distance in the why values is you know, the difference between nine and negative for just 13. So our base is equal to five and our height is equal to 13. So now we have to plug this into our handed Andy equation. We have area is equal to base. That's five times height 13 divided by two. And, ah, if you do that calculation, you're gonna get the area is 32.5 units squared

And the question we have a fear. It's uh right triangle, the whole figure. So we have to find out the area of this shaded triangular region. So, first of all, let's find out the area of this whole the big triangle, which is which base is we can see B. He calls to 12 plus six, which will be 18 for the whole triangle. So the height will be from here to hear. So for this whole triangle, the height will be eight plus five. He goes to 13. So the area for the whole triangle will be yeah, half times base times height. So it will be have into base which is 18 and how it will be 13. So after calculating that, we'll get the value which is one 117. So this is the area of this whole triangle. So if we minus the non shredded apart from the whole triangle, we will get the area for the Syrian regime. So here we can say the non shaded part is separated from each other. Yeah, there are two right triangles and a rectangle in the non shaded part. So if we find out the value separately for each figures, we can sum the non shaded part and minus from the whole area. So from the it as it is a rectangle. We can say this part will also be six and this part will be fine. So the first triangle which is here, the area for that triangle will be half times based time site. So it will be half times base which is six and height will be eight. So the value will be 24. Yeah. And let's go to the second part which is the direct angle for the rectangle. We can say the area of the rectangle is the multiplication of the two sides. So the area of far this rectangle will be base times height six times five. So it was 30. And for the for this triangle, the area will be halftime based time site. So it will be we are fork rangel too. It will be half time space which is we have we have 12 and high. It will be five. So there we have strangled two will be 30. And here we have area for triangle one. So if we sum the total non shaded part, we'll get that uh totally uh man shredded pork is 30 last 30 last 24. The sun will be last 24 will be 84. So if we miners that from the whole area of the strangle, we'll get developed of the area of the non area of this shaded part. So a shaded part. It will be 107 10 minus 84. Which will be 33 unit square. So yes, that's the answer for this question. Thank you.

We need to find the area of the triangle with overdoses negative 2040 and two threes. We're going to plot the points first, so negative to zero. Then we've got 12340 and then we've got to and then up three. So this is our triangle. So our base of our triangle goes from here to here. So that's 12345 six spaces are height of our triangle is the perpendicular distance to the base, so that's 123 spaces. So the area of our triangle is one half base times height, so area is one half our bases six. Our height is three, so that ends up being nine units square.


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