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(a) If $ a > 0 $, find the area of the surface generated by rotating the loop of the curve $ 3ay^2 = x (a - x)^2 $ about the x-axis.

(b) Find the surface area if the loop is rotated about the y-axis.

a) $$\frac{\pi a^{2}}{3}$$

b)$$\frac{56 \pi \sqrt{3} a^{2}}{45}$$

Applications of Integration

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Missouri State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

The basic question is area of surface generated by rotating a loop off three a y squared equals two x a minus x whole square about XX is and then about why exists so that we have to calculate So we know if this is our let's say X and we have wife and let's assume this is the line week even. And we want to find the surface area off this line after rotating about X axis so we can take a small part off this. Let take the DS, which can be divided into two parts DX and dy y So we can write This s the X Square plus device Square is de square. Now, if we take the x common, we get one plus the were by D X whole square times DX right. And we need the s so we can take root off this. Okay, so this is the d s, what we get. And since we need the surface area, so surface area will be height, which is in this case, is why right? So this is radius times the circumference so we can write the surface area. Small part is to buy are. Which is why times in this length DS so D Yes. Now the total area will be integration off to buy Y d s s route off one plus d y by D x, the X and here excess ranging from starting point to ending point. So let's take that as a to be Okay. So after obtaining this formula now, let's go to where the equation off her loop, which is three a wide square equals two eggs a minus x whole square. Now, if you plot this look, we get a loop like this since at 80 and and X equals to zero. So I took two points. Do we have a blew up like this at Seattle And a So we need the surface area of this loop only and were not bothered about this points. So we have to integrate from zero to a Okay, so we now know our integration. So now we have to find one plus, um, Body X and value of what is already Yeah, right. Okay. So let calculate the you have i d. X. So do you have I d. X off the function? This will be closed toe. So let's differentiate it So six a Why do you have I d x is equals to the by d x off X minus a minus whole square. Right, So, on solving this we get Do you have I d x six a way is equal. This gives X times a minus X times two times minus one plus a minus x whole square times one Right. So we can write this d V by d. X is it calls to minus two x A minus X plus a minus X squared over 68 Why? So we can take a minus x common And here we get minus two x plus a minus X over six a y and this gives us a minus x a minus three x or what? Six a way. This is our do you buy DX? Okay, so we opportunity whereby dx and let's obtain the value of why, since we need 13 our function, so why can be obtained by taking square root off the entire function here? So this will be Route three A. Y is equals to root off X and a minus X, so this gives us why is the root X minus X or route three A. Now let's substitute these values obtained in our equation. So s is integration zero toe A to buy Why, which is root? Tex A minus X over route three A times loot off one plus liver by D X whole square So it is this a minus x a minus three x over six a whole square DX Let's substitute the value off Why this one? So we get zero to a two pi root tex a minus x over route three A times one plus a minus X square a minus three x whole square divided by six square just 36 a square and why square? So why square value is X a minus X whole square upon three A. Okay, so we can cancel sometimes. Here. Okay, so we get three. So this is zero to a to buy and route takes a minus X over route three A. And here we are left with a minus three x square and the no matter we have to well, X a right. So this goes too well, X a plus a minus three x whole square divided away dual X A Okay, so on simplifying this an entitled here we have DX value. Sorry. So this gives us zero to A to buy the route Takes will be canceled by this route. X route will be canceled by this route so we'll be left a minus x on simplifying this We get a plus three x Okay. Sorry will get multiplied and not canceled. So we'll be left with too well in 23 which is 36 36 Rudi's six. So we get six times today. Times a day is a DX. So on integrating dysfunction and simplifying, we get value off. Our surface area is equal. Stood by is square over three not to calculate the volume Sorry surface obtained by revolving about the Y axis. So that is this is our look and we're rotating about why access? So we need DS, which is same as thesis. So we can calculator simplify these ds, as we have obtained here is a plus three x over route off the wealth X A dx. So this was a D s and its surface area will be. So instead of rotating this entire loop, we can rotate the upper part of the loop, so that can double it. So this is two times toe pie eggs times ds, and that is two times two pi x d s s a plus three x over route off too. Well, X a dx. Okay. And here limit off X. We know it is from zero to a great. So if we simplify this, we can write this function as toe by over Route three a integration zero to a It is into X rays to half plus three into x rays to three by two TX So on calculating this we get our surface area is equals toe 56 5 through three is squared over 45 that is over. Answer. Thank you.