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Sketch the region whose signed area is represented by the definite integral, and evaluate the integral: a) f-1t+1dtb) JJ cos(x)dx...

Question

Sketch the region whose signed area is represented by the definite integral, and evaluate the integral: a) f-1t+1dtb) JJ cos(x)dx

Sketch the region whose signed area is represented by the definite integral, and evaluate the integral: a) f-1t+1dt b) JJ cos(x)dx



Answers

Sketch the region whose signed area is represented by the definite integral, and evaluate the integral using an appropriate formula from geometry, where needed. $$ \begin{array}{ll}{\text { (a) } \int_{0}^{3} x d x} & {\text { (b) } \int_{-2}^{-1} x d x} \\ {\text { (c) } \int_{-1}^{4} x d x} & {\text { (d) } \int_{-5}^{5} x d x}\end{array} $$

In this given yourselves part here. Give Undocumented bill can be reprinted by the separatists and given like that. So period Wrangler areas of a physical one back to base into height. And I, it's want and based to Kota video is one of our build abortion. The different, particularly that keeping by the Christian here and to find the area did your crap idioms because one by two to somewhat parallel size. But like the height that is according to. And the Pakistan would send a definite integral, obviously affronted by the idea given by a sensitive, you know, die except Cecil. We sort recruiters minus a so it is a triangle of areas which will be minus one my proof until baste baste it one and height is one way to Don said his minus one Matthew. Similarly, for party of the question the garbage given by district land an area off the definitive given by a one year two. So everyone will be quantity and it will be a good German boat. Anglers, Yes, but I have only sequel one an idea to pull one back. So now you said it perfectly

Let us value this integral geometrically. So let's see the first one. So the first one is integral. 05 two d x. The function here is a for fax is equal to do, which is a constant function. And what are the limits off integration? Zero to fight, so it simply looks like this. So this is my excitement axis zero and five easier. The constant is too, so it's simply constant, too. Something to go fight. Basically, it's a rectangle and according toe, the geometric meaning off definite integral it represents assigned area. But now this area is about X axis. So the science area is positive and what will be the area off rectangle, its victim to length. So this distance is five units, but is this distance is two units, So my integral 052 d x will be 500 to between two length. So tennis sans that's it. Let's see. Zero toe by integration off cause affects DX. That's a draw. The graph off course X from zero to pipe. So this is my zero. This is my pie. It crosses eggs access by by two with a maximum value of one. So it looks like this. Yeah. So this is my graph. So from zero to buy very interest rate. So this is my graph. So this is a pipe by symmetry. This area it's canceled with this area because the definite integral represents the signed area. So there will be an algae by examination, not Absolut summation. Unsigned area is different. Signed areas, different unsigned area. If I need from zero to buy cause affects, then I should double this value. But here one area symmetrical is about the X axis. The other area is symmetrically below the X axis. But the areas are equal by symmetry, so that gets cancel each other. So the need area zero. So that means my internal zero to buy cause affects the X simply zero. That's it we have done. Let's see the next function more to x minus three from minus one toe So minus wantedto models off two X minus three D x Martinez off two x minus tree. It will be a reshaped call the critical point at three by two. So my model is off to explain street coughs. He looks like this and this is my three by two so my to win is somewhere here, so my function is worked. Models off two X minus three. Mannix zero. What is my answer? Three this height history. But the next is minus one. It will be minus two minus three, which is five. So that means So this is five. And this is my minus one. So basically and afford to write a photo. Also to will be models off doing 2 to 4 minus three, which is one so one will be somewhere here. Yeah. So this is what we need. Basically, Yeah. So one in some area, roughly. I'm drawing the graph. Yeah. So basically, what we need is the area under this girl from minus one to do so minus 1 to 2 covers this triangle and three by two, cause this strangle. All right. So basically, we need to add areas off two triangles. One triangle basis three by two minus off minus one. So this basis, the land if I buy two, where is this basis? Off the land to minus three by two, which is one by two. And this strangles hide is obviously one, because for minus three models of his one but as this Chinese Heidi's fight because model is off minus two minus three is five. So what is the some of the areas with these two triangles? What we get is half off five by two into five, less half off, one by two into one. So it's 25 by four, plus one by 4 26 by four, which is 13 by two. So this is the answer, and we're done. Let's see the last one minus 1 to 1 Ruben minus Exit square. If you consider a function F of X is rude one minus X squared, which is basically what he could do. When you square on both sides, you'll get an equation like this, and this represents a circle the entire circuit. 3 60 degrees. But now what is the difference you see here when, when I write wise Route One Minds Square, my wife is always positive because, as Chris functions are concerned, anything any square root should be taken as positive. So that means my circle should be only the upper half, so it should be only this part from minus Wanda, because this circle is actually this one. It's a subtle with central region and really is one. But now we need Why is positive. So my arc should be only about exactly is only this part. So it should be a same, isn't it? Now what were the area off this by integration? It's a positive answer because it is a public taxes. So minus 11 road one minus X squared. The X is simply the area off semi circle, which is half off by our square. But what is the Radius one? So it's bye bye to that's it. So we are done.

Let's value this integral geometrically. The first one is minus 10 to minus y 60 x, so that means we have my functions. Six limits off integration are from minus 10 to minus five, but the high 86 because the constant function he's staying at a level of six throughout its journey. But our limits off integration only from minus 10 to minus way. So I need only this part of the area. This area is about exacts is so the definite integral should be positive. And what does this area represented? Area of rectangle with Haider's six units and what is the best? The difference? Off minus 10 and minus place five rooms. So my minus tinto minus phi 60 x the integration will be simply 16 to 5 30 and we are done. Let's go to the second question. It's minus by by 32 Bye bye. Three Sinek sticks. Let's draw the graph off cynics from minus pi by three toe by by three. So minus by my trees over here by mid eighties or here. So my graph looks something like this. So basically it goes like this. Basically, this is minus pi by two This is by by two maximum Occurs it Bye bye to on minimum occurs set minus by by two. So this is my minus one And this is my one. But we're interested only from minus private. Reto Private Re basically I need this area in this area. The net off this area by symmetry One area is about exacts is one area is below exactness and border off equal area magnitude is seem because of symmetry because science is in our heart function. Every art function is symmetric in oppose it Quadrants. Very interesting property. So what will be the total signed area? If this area is a, this area is minus a good gets canceled a plus minus zero. So the answer for this individual is simply zero and we're done integral 0 to 3 modelos off X minds to DX. Let's draw the graph of more X minus two. So this is more. It's minds to graph. It's right from 0340 is here zero. The height of the function is to because if affects is more X minus two, what will be a 40 more zero minus two, which is two. But what will be a three threes more three minus two, which is one so a three. It is simply one. So that means I need this area, plus this area because we're interested in +03 This one is a triangle. Off based two units. Height. Two units. This one is a triangle off base. One unit height, one unit. So let's at both the areas strangles half base into height, so have to into his first frankly area. Let's have one into one second trying Lydia. So it's four by two that is two plus one by two. So basically the answer is five by two. That's it. So we are done. Let's see the last wish in the 0 to 2 Rudolf four minus X squared The X Remember why I called you Root for minus X squared presents a person me circle Because when you square on both sides and get a certainly equation X squared plus y squared equals four, which looks something like this, its central origin nine releases to. But when you draw the graph off by equal Route four minus X square, why is always positive? Because Squire out of four minus X squared should always be no negative. That means the circle should be circular. Part should be about X axis. That means I should take my semi circle. Is that region as they come? So this is my semi circle from minus two from zero toe comma 00 We're interested from zero toe to only. So we're interested in the quarter circle area? Not even so musically. Simply quarter circle area. So what is the idea of quarters again? 1/4 off by our square and raises too. So it's 1/4 off pine to to square. So basically my answer for this interior list, but to answer this pie, we're done.

Okay, so we're going to start by drawing a graph and the question we're dealing with ihs Absolute value of X, and that creates this be shaped craft. We are looking at the area under the graph. That's what the integral means from negative, too. Tow three. So negative too. My Why value is going to be too. And at three, my my value is gonna be three. So we have two triangles here. This one has a base of two in the height of two. This one has a base of three in the high of three. The area of a triangle is 1/2 based on site, and we're gonna have to add or two areas together. So I'm gonna have 1/2 basis to times of hide up to plus one, huh? Basic three times the height of 30. And again, they're both above the X axis. Other positive. This is gonna give me four halfs and this will give me nine over two or nine. House. When I had those together, you're left with 13 over too. Or six point thought. How's your Yeah,


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