show-that-the-area-of-the-ellipse-in-the-accompanying-figure-is-pi-a-b-text-hint-text-use-a-formula-

Question

Show that the area of the ellipse in the accompanying figure
is $\pi a b .[	ext {Hint} 	ext { : Use a formula from geometry. }]$

Carson Merrill · Accepted Answer

it was probably when I show that the area of an ellipse is pie A. B. So let&#x27;s keep in mind that our lips would be X squared over. Let&#x27;s say um One plus y squared over. We&#x27;ll say four. It has to be a different value. If it&#x27;s the same value will end up getting a circle equals one. So this gives us an ellipse and we see that by changing this, change the shape of it, this is going to be a squared and this is going to be b squared. So our area is going to be pie baby. one way we could show this with integral is by solving for why. Um If you solve for y you&#x27;ll end up getting this equation right here. Subtract this multiplied by four. Um and then take the square root. When we take that square root, we&#x27;ll have half of the area that we can find it and then we just double that.

Angela Guo · Answer

in this question were given this equation, which is an equation for an ellipse that were asked to show that the area of the celebs is equal to pi times eight times beat. So let&#x27;s draw the lips first. It looks something like this going from negative, eh? A. And what is going from they could be to be. So what we&#x27;re gonna do is we&#x27;re going to I&#x27;m sold the secret for y fished. So we&#x27;re going to get why? I swear it is equal to one minus X squared a squared times. So which means why is going to be this groubert of that? So the area of the alleged CZ, all that eh, It&#x27;s going to be the Inter girl from Negative, eh? A of the bomb? Uh, you square root, huh? He squared finds one minus squared older Hey, squared D X, And then we have two times that by two. Because the square root, we&#x27;re only taking the positive script route. So? So when we do this will only find the area above the X axis. So that&#x27;s going to times, too, because I need the area under these as well. So now we&#x27;re going to do a substitution. So we&#x27;re going toe. Let view the equal to X over a then d u is going to be equal to one over a the ex and go forget wills and changed our change, Our limits integration. So I knew lower limit that is going to be equal to negative day over a some negative one and a new upper limit that&#x27;s going to be positive. A over a CZ likes going to see one. So now this is going to be equal to two. Um, Well, factor out the square root of these great witches Be so we&#x27;ll pull it to the front and this is now then trouble from negative one one of discovered of one minus you squared. And in D X, that is just equal toe. Hey, tons, please. So again, we will pull the way to the front. So that&#x27;s two a and then into trouble. Negative. One toe, one of one minus you scared you. So if we look here, um, this is just the one line issue squared, Um, that looks like that&#x27;s so this is actually you&#x27;re a semi circle. So which means that we have two times that that is going to be the, um you know, the area of the unit circle going from the gate of 1 to 1. So which means it&#x27;s going to be a Times B times and that it&#x27;s just area of a unit circle and that is going to be equal to pi, So our area is going to be a Times B times.

Suman Saurav Thakur · Answer

Yeah. This is number 41 in which we need to show that the Ellipse with semi major axis a and say my nexus B has an area equal to by a B. Okay, so general question will be access square by square plus y Squire by B squared equal to one. So let us draw the ellipse and let us, uh, assume the major exist is a long X success, right? Roughly, the lips should look like this. Yeah, so we can easily see that ellipse are being divided into four equal parts 123 and four. And if he just, uh, get the area of this part, anyone off this? So let us say we re finding area of this part a a and the multiply with four. So we will be getting the Haley off area of hold the lips completely. Lips. Okay. So, uh, this this point is a comma zero. Okay, so let us take a strip, which is vertical strip with thickness d x. So we have to find the value off. Why? So from here, we&#x27;ll be getting why Squire by B squared equal to one minus access choir by a square So why square by the scar will be equal to a square minus excess square under its requirements X squared by a square. So why square will be equal toe be a squired by a Squire A square minus x squared So why will be equal toe? It will be good plus Minebea by but we&#x27;re talking about quadrant one. So let us take only plus positive value. So be by a a Squire mine Sex Squire under route No. We will form an integration total area, though First of all, we&#x27;re finding area one that is on leave for first quadrant. One quarter of the lips, which will be, uh, the strip will move from zero toe A. So let us right the limits as zero to a be by a a Squire minus taxes choir minus zero because at the base the value of y zero And here at the top somewhere here value of why general value Is this what we have got? And the X is the thickness. So we have to integrate. We have to integrate zero to a be by a Squire minus x squared d x. So just as Bieber is common so let us take be by a outside integration becomes zero to a a square minus X square DX. Now we have basic formula for integration this type of integration. So we&#x27;ll be writing be by a zero to a formula sees the integration is will be acts by two a square minus X square on the route Bless a square where to sine inverse x by a There will be no constant of integration as this is a definite in deal. Okay, no, be by a Now let&#x27;s start plugging the parliament first that is really plugging in a in place effects. So this will be a by two a Squire minus a square and the route plus a square by two Sign in verse A by a minus. We will now be plugging in a zero in place of x zero plus a square by to sign bar zero. So this will be be by a this will will become zero as your requirements is quite zero plus is quite by to sign in verse one. So sign was one is always bye bye to so be by a a Squire by two and two by way too. So this will get canceled out, so we&#x27;ll be getting a beep I by four. So area off one quadrant is equal to buy a beer before. So total area off their lips will be four into a That is four into by a B Before this figured, cancel out. So total area will be equal toe by a B. This is broad. Thank you.

Sriram Soundarrajan · Answer

you&#x27;re going to start. Probably number 28. Age off the islets. Rhenish to fight ill It is Goodbye X squared plus X squared plus y squared by B squared equals one. So area equals every major. Oh, look. Oh, the geek Excessive. Ricardo. They, uh, ghost pickle and where he was Be a saying, Dedeaux, we have us where equals one for are really because to what early? It&#x27;s like, uh is a subset id our coma bigger, isn&#x27;t that zero is less than every personal detail As then. I request your group by zero. Less than our egos toe are less than or because to what? So they&#x27;re coping off hard. Come on, Dig is a quarter. Oh, x by Know what a growth Diduck The works by ditto minus stale ST Peter It&#x27;s be saying they don&#x27;t and be, uh, cause they don&#x27;t. So which comes to be it be Are goes where the dark plus same square. Dedeaux, this is a Porto A b oh, food me is he glared number a little over our our daily. This is a total nobody doesn&#x27;t over. Our is a good, you know, Hot goma d Don&#x27;t the, uh and the editor. It is a coastal zero, do, but And Theodore Bill. Fine. It be. Are the, uh they didn&#x27;t go. I was using a photo group by E into. Off Where by do is he is or don&#x27;t apply. What is your big getting at best by mm.

Ankit Gupta · Answer

we see from the class, it does give it God B e Clewiston, which is equal to the ladies off The soca on the area of the circle is equal to by our square, but as radius is be so it&#x27;s by be split. And it&#x27;s also given this is equal to area of the ellipse divided by two on the area of the Ellipse is given us by a bee. So this means by bees for equals, baby upon to this gives us a equals Toby as being equal stand a equals 20 on length off the major access is equal to a which is equal to 40.

Georgiann Andersen · Answer

Okay, that&#x27;s this problem. 38. And it says that you know that the area of a circle is pi r squared. If you have any clips, that&#x27;s a horrible circle. Um, which goes like this, and this is B. This is B. This is a and this is a just find the formula for the area of the eclipse shown below. Look, you have a circle, you know that these are all the same measure. So this would be in a This would be an A on A on A So the radius is a So we know here that the circle I like it said would be pi a square. Now, here with the cliffs, you would have to take pie. And instead of having a squared, you&#x27;re gonna have to take one that one of the right radius times the other one. So you&#x27;re gonna have to take a times be. So that&#x27;s what the formula iss horny clubs

Problem

Writing Suppose that $f$ and $g$ are continuous o…

Problem 50 Easy Difficulty

Show that the area of the ellipse in the accompanying figure
is $\pi a b .[\text {Hint} \text { : Use a formula from geometry. }]$

Answer

$A=\pi a b$

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$A=\pi a b$

Calculus Early Transcendentals

Catherine R.

Anna Marie V.

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Integration Review - Intro

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Howard Anton, Irl Bivens, Stephen DavisCalculus Early Transcendentals

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Joseph L.

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