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Determine the area between the parametric curve and the y-axisin the first quadrant for the curve given by𝑥=1−𝑡2 and 𝑦=𝑡3+1.So...

Question

Determine the area between the parametric curve and the y-axisin the first quadrant for the curve given by𝑥=1−𝑡2 and 𝑦=𝑡3+1.Solve and explain how to get the interval.

Determine the area between the parametric curve and the y-axis in the first quadrant for the curve given by 𝑥=1−𝑡2 and 𝑦=𝑡3+1. Solve and explain how to get the interval.



Answers

Sketch the curve and find the total area between the curve and the given interval on the $x$ -axis. $$ y=\sin x ;[0,3 \pi / 2] $$

Gonna find this integral here from 0 to 3. So the sketch of this just look like this, right, says you're a one here, It's got this area here in this area underneath here. Okay, so trying to do here if we're trying to find the total area, so going to do from 0 to 1 of negative X squared plus one, Then from 1 to 3 of X squared minus one. Okay, so It's going to give us negative, execute over three Plus X from 0 to 1 That's next to the 3rd For 3 -1 123 So that gives us negative one third plus one plus zero here Plus 27/3 -3 -1 3rd plus one. Simplifying all this down, you get 22/3 units squared. And so that's the answer there.

So this question were given that y equals either the power of negative six sacks and x axis. So the area in the first quadratic between wine the x axis we can right as this interval. Yeah. So now we can make this interval into a limit limit is he goes to infinity. Uh, t zero. You know the power to your six Anstey, which we can then rewrite. This portion is negative. One over 60 the power of negative acts. This is Europe. So up here and now we have the limit I see goes to infinity one sex minus 16 year. The power negative Cheap. Which one we evaluate is gonna give us 16 which means the area and this washing is one sex.

Problem. We're looking for the area in between X minus y cubed equals zero and X minus Y equals zero. So I've done that graph already. The X minus y cubed equals Here is the blue curve. And why equals X or X minus y equals zero is the green curve. Ah, we're looking for the area. So we there we see that there are two, um, regions, but they're symmetrical. So I'm just going to find the area, uh, in between 011 So the blue curve would be the upper curve, and the green curve would be the lower one. And if I multiply the area by two, that'll give us the overall region. So area here is equal to the two times the integral from 0 to 1. Um, we are actually going to do this with respect to why? Because that'll give us nice polynomial is to work with, um So instead of doing, the blue curve is the upper one. We're gonna do green as the upper one, the right most and blue as the left most. So that's going to give us, uh, why and we're gonna take away. Why Cubed? That's the left most curb. And this is with respect, Toe. Why do you? Why so anti derivative? That's gonna be one over to Why? Squared minus mourn over four. Why? To the fore evaluated from 0 to 1. Evaluating at zero is gonna zero up. So no point. We'll just put one in there. This is one over two minus one over four, and this is two times. Ah, half minus 1/4 is 1/4. It's a times the two would give us 1/2 is the overall area.

Um so next case we have up record. Why equal Dax? Um, between exit zero Ebola. At which point it equals one as well as the curve one of Rex square. Then between one and two we have the upper curve being one of Rex squared, which decreases from 1 to 1/4 from Mexico under two as like a sex increases from one or two. And of course, the reason we stopped to is that the line X people too? And we're integrating. Ah, again from er the X axis, which is wise, you know, teach Kirk the two curves. I'll see which of the two curves still together. Then we integrate X windshields explored by by two. Well, this is just between zero and one. Animals went to grade What Obrecht squared Would shields minus from brags this time by winning between one. It eso this yields half that's here was 1/2 Ah, less 1/2 0 less, um, minus one s. So we're left with is less my son wishes is possible. Um, actually, uh, we we can visualize not about drilling or just mentally visualizing, actually. So we have that. Why we have that x is one over X. Furman X Cube is one which has two complex solutions as well as the social nexus. Once we integrate between X one and X is too, uh, those values we've already seen that y equals X is greater than I could At least Weikel 24 x squared. So we integrate X minus one over X squared from exes. Wonder too, that yields. And you Derivative exported by by two plus one of Rex. Yeah, in this whole, uh, derivative is evaluated between X is one where the 2% is act nexus to correspond to the vertical line down in region. On a really good question, eso this yields afforded by by two is two plus 1/2 is five halves. A less 1/2 is to less units. One book, um, drunk so more carefully. In some cases, it's best to do so So we get about this blue line, be it access to you can let the green nine b y equal tax and, uh, contempt to sketch. For instance, exes to we will have 1/4 of the value of one of our square so you can attend to scratch something when the phone runs. And so, indeed, we're bounded by the land Money quacks should screen the line exes to which is Boo Curve. I was expired, which is black. The X axis, which is also black. Uh, Sophie, Like we've been shaved in red. What we read, um, integration. And again before the two Kurds country sect at X is one region squid are integral to turntables was pursued initially rather than the area bounded between. Why cause X y go long? Rex Ryan Exes to Mitch. We can like bacon. Hash. Instance. Injuring it was computer in the second attempt. But again, it's the red Hannah Sean that we're interested in. Um, no. Again, this means that we integrate eggs from 0 to 1 she gives that scored God, thank you. Into this. We had the interval of what? Greg Square, which is my sort of Max. This is evaluated between one and two again. This gives us 1/2 Ah, subtract when half which is zero and then subtract. But this one so just one. But interestingly, we observe that theory of your direct triangle of sideline to and high too is just four divided by two, which is to that implies that the green area has area one. I wish we could bid on the second attempt, but in the area one time in the first attempt actually was correct. We confirmed by dry note a mental image of the situation. And so this just in decades that in this case, um, the answer provided it's actually correct. I mean, not very well over six, and the correct answer is just indeed one, as was found on the initial attempt but which that's strong illustrates the two attempts and a clear picture by the Craig answer is long one of the six, and that completes this question solution.


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