Hello everyone. Welcome to the session of application of multiple integrals part 2. This is Swati Nikam, Assistant Professor, Department of Humanities and Sciences, Valchan Institute of Technology, Sulapur. At the end of this session, student will be able to find area of region bounded by curves by using double integration. Before we proceed further, evaluate this simple integral by pausing your video for 2 minutes. Integration 0 to 1, inner integration 0 to y, x, y, dx into dy. I hope you got your answer. Late, capital I is equal to integration 0 to 1, inner integration 0 to y, x, y, dx into dy. As the inner integration have limits in terms of y, so let us integrate with respect to x first, which is equal to integration 0 to 1, integration of x is x square by 2, limits 0 to y, this y we have to keep as it is into dy, which is equal to integration 0 to 1. Now, put upper and lower limit in place of this x, so y square by 2 minus 0, outside the bracket y dy, which is equal to integration 0 to 1, this y square into y is y cube divided by 2 into dy, which is equal to, now integrate this term, integration of y cube is y raise to 4 upon 4 into 2 is 8, limits of this integration 0 to 1. So, putting limits, we will get it as equal to 1 by 8. So, we have evaluated a double integration and got this answer. Let us solve an example, example number 1. Find by double integration, the area bounded by the parabola y square is equal to 4x and the line y is equal to 2x minus 4. Answer, y square is equal to 4x is the parabola symmetric to x axis, let us call it as 1 and the line y is equal to 2x minus 4. In order to plot this line, we will change it and let us write down its equivalent form as 2x minus y is equal to 4. Divide this equation throughout by 4, hence we get x by 2 minus y by 4 is equal to 1. Let us call it as 2. Now, this is the line, it passes through the point 2 comma minus 4. Let us draw these two curves on the graph, so that we will get the required region of integration. After plotting these two curves, region AOB is the required region bounded by the curve 1 and 2. To find coordinates of the point A and B which are points of intersection, we have to solve equation 1 and 2. So, to find the points of intersection A and B, let us solve y square is equal to 4x and y is equal to 2x minus 4 simultaneously. Now, let us square this equation. Therefore, y square is equal to bracket 2x minus 4 whole square. But y square also represent 4x, that is why y square is equal to 4x is equal to bracket 2x minus 4 whole square. Therefore, y square is equal to 4x is equal to open this bracket by using A minus B whole square formula. So, it is 4x square minus 16x plus 16. Shift 4x to RHS, hence 4x square minus 16x plus 16 minus 4x is equal to 0. After simplification, 4x square minus 20x plus 16 is equal to 0. Divide throughout by 4 and therefore, x square minus 5x plus 4 is equal to 0, which is simple quadratic equation in terms of x. After simplification, it has roots x equal to 4 and x is equal to 1. Now, when x is equal to 4 and y is equal to 2x minus 4, this together implies that y has value y equal to 2 into substitute value of x as 4, 4 minus 4, that is 8 minus 4 is equal to 4. Therefore, point A has coordinates 4 comma 4. Similarly, when x is equal to 1 and y is equal to 2x minus 4, this together implies that y has value y equal to 2 into value of x is 1 now minus 4, that is 2 minus 4 equal to minus 2. Hence, point B has coordinates 1 comma minus 2. Now consider a strip parallel to x axis, say PQ, so that with the help of this strip we can write the limits of this region for integration. To find limits of integration on this strip, x where is from. So, already we have studied about finding the limits from the region that whenever the strip is horizontal, then we have to consider limit from left end to right end and hence x where is from at the left end x is equal to y square by 4, 2 at the right end x is equal to y plus 4 by 2 and y where is from. Now, y has constant limit at the bottom y is equal to minus 2 to y is equal to 4. Therefore, the required area A is represented as outer integration minus 2 to 4, inner integration y square by 4 to y plus 4 divided by 2 into dx dy which is equal to integration minus 2 to 4 of as the limits are in terms of y. So, integrate with respect to x, integration is x, limits y square by 4 to y plus 4 by 2 into dy which is equal to integration minus 2 to 4. Now, let us write down limits in place of x. So, upper limit y plus 4 by 2 minus to lower limit y square by 4 into dy which is again equal to, let us find out the integration now. Integration of y by 2 is y square by 4 plus integration of 4 by 2 that is 2 is 2 into y minus integration of y square by 4 is y cube by 12, limits of this integration minus 2 to 4. So, here A is equal to bracket y square by 4 plus 2 y minus y cube by 12 outside the bracket limits minus 2 to 4 that is equal to in place of y we will substitute upper limit and lower limit. So, that is equal to 4 square by 4 plus 2 into 4 minus 4 cube by 12 minus now the lower limit minus 2 square by 4 minus 2 into minus 2 plus minus 2 cube by 12 which is equal to 4 square by 4 is 4 plus 4 into 2 8 minus 64 by 12 minus 4 by 4 plus 4 minus 8 by 12 which is equal to. Now, let us collect all the integers in one bracket 4 plus 8 minus 1 plus 4 minus let us have minus sign common from these two rational numbers and in another group it is 64 by 12 plus 8 by 12 which is equal to 15 minus 6 that is 9. Therefore, area A is equal to 9. I have referred higher engineering mathematics by B.S. Graeval for creation of this video. Thank you so much for watching this video and have a happy learning.


Welcome back to my channel. I hope you're all good. Today. We're going to be chatting about my January rapper. I Don't want to do it Before we get into it, this is my first video. I'm filming since I hit 10k I cannot tell you how. Thanks. I am like my brain can't really wrap around it Like my brain doesn't really get it that number of people. Thank you so much. Thank you so much for 10k Thank you so much for all your continued support I actually want to do a 10k Q&A to celebrate this. Yeah, if you have any questions for 10k Q&A Let me know down below and I'll include them in the video. Now on a sadder note I was in a bit of a slump in January.  Some of you like caught on I don't think I've uploaded that many vlogs in January. That's because I haven't read a lot I mean I read eight books. It's not terrible. I'm hoping in my head I'll read like ten-ish in February but considering it's the third today when I'm filming this And I haven't started a book yet That's not great. It's made a fool of me The way I do my wrap-ups is we're gonna talk about all my reading statistics For the stats people of you out there and then we'll talk about all the books I read or eight of them I will put a timestamp down below if you are not interested in hearing the stats But I love the stats I use Ali from hardback hoarders spreadsheet to track everything I read a total of three thousand and forty three pages this month Which is I think on average like 98 pages per day my rule of thumb is that like minimum I like to be above a hundred pages a day on average So it's not like quite where I want to be but it's not that far off It's like two less pages a day. Hopefully in February we can rectify that It's not realistic It's just not realistic But I'm trying not to be too hard on myself because uni is very busy and I I feel very overwhelmed By the way, what do we think of this jumper? Is it ugly enough? It's I'm buying those are clothes for my birthday. That's just been like that's what I asked for was money for clothes and I love it so much, but I know it's very ugly, but that's what we go for star rating I gave two five stars, but one of them was a reread so doesn't really count and one was the end of a series So I feel like I am craving a surprise five star I want to be hit in the face by a five star like I want I really want a surprise five stars I hope we will get that next month and one four point five star two four stars one three point five one three and one Two so my average rating is like a three point eight, which I think it's fine Like it's fine. I think that's pretty much what my like overall forever average rating is like a three point eight So it says here are seven physical books and one audio book, but I think the most books I read along with the audio book and the physical so that's not quite true But I did tell a bit of a lie there I actually did a ton of audio book reading this month and it got me through the month And like I wouldn't even have read eight books if I hadn't read some audio books in terms of genre I read one contemporary four fantasy one historical one horror one mystery half of the books I read were fantasy books Fantasy is always I would say like one of my highest-read genres just because like it's so popular But I'm kind of like not in the mood like I feel like I need a month's break from fantasy I want more like mysteries and stuff right now half of the books I read this month I categorize its own voices be that to do with race or Sexuality or something along those lines and then in terms of author race I read from two black authors to Asian authors and four white authors So in my reading goals for the year a big ghost me this year is to make sure that the kind of percentage of white authors Is 50% or below ideally below below, but I think one month's worth of reading isn't the best Indicator because again, it will be very much determined by the videos that I do and things like wrapped up Or when I have no choice on choosing a book It does make it more difficult to ensure that because I cut yeah I can't ensure that you know like two out of three of the videos I'm doing this month for vlogs. I am out of control in what I read So only that one video can I like make sure that I am reading Authors of color so it doesn't make it more difficult But I'm happy with 50% in one month like as long You know it's a good starting point and I read from one day be author five authors I read before and two authors were new to me seven were part of a series one was a standalone. That's pretty crazy I read a lot of series books this month. I think that's all of our stats So let's get into talking about the books the first book I read this month I read on the first day of the year and that is when the tiger came down the mountain by me By so this is the second book in the singing hill cycle and these are really short the values Beautiful beautiful colors beautiful writing all about this cleric called She who is a non-binary main character, which I absolutely love to see I don't think that we read or we have enough non-binary main characters in fiction They are a cleric who goes from like town to town area to area and they have to write things down and record human history I would say and in this they are they and their companion are cornered by these tigers who make them tell this story and they go back and forth with the tigers and the tigers correcting them on the story of this tiger who fell in love with a scholar this sapphic Relationship that is kind of like this folklore tale almost I have preferred it to the first book I mean I love the first book But I love this one even more because of that story telling element where like you'd hear the story one way and then you'd hear The story another way and I think that was just really interesting way of examining like folklore and mythology but like old tales You know like tales to the grapevine and how the truth can be like not twisted But you know adjusted as the story goes through many hands and many mouths and Nevo's writing is just absolutely beautiful like it's just such gorgeous Writing that is really emotionally impactful and just really I feel like cuts to the core. I absolutely loved it I would really recommend this book series. I think it's just like Stunning it's absolutely stunning and I cannot wait to read everything that Nevo puts out I can't wait for her first full-length book coming out this year The next book I read this month was a reread and it was the girl in the tower by Catherine Arden This was five stars obviously of course of course of course of course a hundred percent of course a hundred percent of course In this series we follow Vassia from birth like we follow her from before she's born to this young woman who can communicate with magic and spirits and The winter king and it's very magical and set in old Russia in winter and it's snowy and cozy That isn't a very new environment And it's very interesting to see how she has to negotiate it and her role as a woman and gender Differently how she has to use her magic and understanding of magic to get through the situation. It's just very beautiful I it reads like a fairy tale This is like one of the only books I've ever found that reads like an old Russian fairy tale Like it's in the style that it's written and it just feels like this warm hug I just love this series so much if you like Fantastical stories with strong young women. I loved it, but it was always gonna be a five star So like I feel like it doesn't really count because I didn't read this for the first time This is like an old love then next I read the fifth season by N. K. Jemisin I gave this 4.5 stars. I was absolutely shook So basically I read this for my Booktubers give me reading prompts video and Aaron gave me the prompt to read the most intimidating book on my TVR And this is app. This was it like I was so scared of this book. It was skate It was scare we It was really scary I would not have read this for like a year without Aaron basically forcing me to in this we follow three Girls slash women we follow one young girl who has just learned that she has the power of Origins which is people who have power over the earth and like the energy within the earth So she's just learned that she has that power We follow a young woman as she is on this mission with the kind of company or organization Who train origins because origins are very feared of in society because of their power They're very much shunned and so like the only place they can work really is within this company And she is on a mission with this man that she has to have a child with and then the last one We follow is this woman who has been hiding her powers for many years And she finds out that her son has been murdered by her husband The fifth season is happening and seasons is where there's like this event caused by an origin and The world is ending and so we just see all these different women Negotiating the world around them. It's very confusing. I Haven't got the slightest idea For the first third. I was like girl. I have no idea what's going on I like didn't love the first third because I was so confused like I was I was stumped But I feel like something happens towards the end of the first third where you start to like make Connections and understand this complex world If you started it and like you're not enjoying at the start like just take it slow I would recommend the audiobook. It definitely helped me through that first bit Just like ease yourself into this because I am felt so dumb I felt so dumb it was this incredibly complex clever fantasy And I can't wait to read the next two books in the series I feel like I need to read them soon to like actually understand what's going on But also I'm in a slump So I feel like maybe not February because February I need to get out of this slump Maybe sometime after that we'll read it because I just want to like get through this series Why I still remember vaguely what happens the next I read the year of the witching by Alexis Henderson And I gave this four stars now This could have been a one point of five star the contrary to the fifth season where it was the first third I didn't really like I didn't really like the last third of this I just found it a bit boring like the last third but anyway in this we're following Emmanuel who lives in this cult And I just love cults like the vibe is just perfection So interesting this is disgusting. I like it though basically years and years ago There are these four witches who bought plague on the on the land And they were driven out by the prophet and the prophet is like the head of the church the head of the cult Emmanuel gets given her dead mother's diary by these mysterious figures in the woods in the dark wood Which is like a place that you're not supposed to go it kind of goes on from there It is the story of this like toxic cult which is back-sepping and betrayal It's a story of Emmanuel negotiating her position in this cult as she is mixed race and Her father was killed for having a relationship with her mother And so she's kind of like always on the outskirts It's kind of rejected by the cult in many ways because she is mixed race it is horror But it's very light horror so like I would say I read this for Aaron's book club the busybee book club We talked about how like it's got elements of horror elements of historical elements of fantasy And we said like if you are a veteran like you read a lot of fantasy But you want to get into one of the other genres like horror. This is a really great way to do that and vice versa So if there's any of those genres you want to dip your toe in but still feel familiar because you read the other genres This is a great way to do that because it's kind of light in all its elements of those things It's light and it's fantasy. It's light and it's historical It's light and it's horror. It's a good mix of them all taking a little bit of them all And so I feel like it's a great starting point in so many different genres for that reason There was some really interesting Relationships in here. I thought the writing was great the vibes the vibes is what carried this book set of the book the actors the storyline plots literally I am obsessed and I long to be in this book Please make my dreams come true someone that's watching this I read the whole thing to the far cry 5 soundtrack don't tell anyone. I Love the far cry 5 soundtrack. I don't know if I can play it I don't think I'm gonna risk it because of copyright, but it's like I'm gonna read a ton more cult books this year I've only got a few lined up because it's just a bit of an obsession like I just if you have any good Recommendations, let me know. I want all the cult books recommendations after I read that one Maybe I should do a whole video on it Let me know great book. It really excited for the second one in the series. I have no idea where it's gonna go I've no idea where it can go. I'm terrified that I'm excited. Okay, and then the next book I read was this time will be different by Misa Suguria. This is a really cute Contemporary I read it by audiobook and I really did enjoy it. I gave it 3.5 stars It was close to being a four, but the ending was just a bit I wouldn't say disappointing just a bit meh, you know like the ending was like We went this whole way for that But in this our main character works in her family flower shop Her I think it's her grandfather had to sell it to this rich white American guy back in the day For a very small figure and then bought it back for an extortionate amount And then her mother who doesn't really run the flower shop it's more her and her aunt her mother wants to sell it to the same family who had bought it and resold it for an extortionate amount and Obviously they don't want to do this and so it's the story of anti-Asian racism in the US and the history of that and A bit of activism and fighting for what you believe in but it's also like a cute romance as well So it kind of balances those two aspects, which I always really like from my contemporaries like my young adult Contemporaries, I like them to have like a serious topic that they're discussing But also have like the cute romance side or just have a serious topic But I don't mind it when they really have the both together. I thought this was really nice actually I thought it was well written. I thought it was fun. I thought it was quick fast-paced I really loved what I learned about like Asian history throughout America I would recommend this if you're wanting like a quick audiobook on script for something That's like very easy to consume because that's what I needed this month I would say though that the ending it felt a bit anticlimactic And it felt like it didn't fit like the realizations at the ending Didn't fit with the tone or the rest of the book and what it felt like we'd been building up to We were on our way to being taught a certain lesson and the lessons switched at the end if that makes sense But other than that, I really enjoyed it like it was gonna be a four star But I just like downvote it half a star because of the end. Okay. I I don't want to talk about this. I Don't I don't want to talk about it like I don't I don't I do not want to comment Why? Why? The next book I read this month was the sinister mystery of the men's Morrison girl by Theo Doragos I gave this five stars. I loved it. I loved it very much. I loved it so much This series is like one of my favorite series ever and I'm not okay. Like I'm not okay that it's over I'm not feeling it that doesn't fit with my fantasy. No, I'm not here for it I'm sorry. Like it's just not I'm really sad and I said in the vlog that I wasn't capable of like making up thoughts and that I'd have a more succinct Feelings about it in my wrap-up not happening. I Still don't know how to articulate my love for this book So in this series, we are following these girls who are like daughters of famous Victorian men So we have Mary Jekyll and Dr. Hyde.  We have Catherine Monroe We have Justine Frankenstein. The men in their lives had been members of the society called the Société de Dépénés And this series is kind of about investigating the society and uncovering things about it But this book doesn't really cut like focus on that. It more focuses on So in this Sherlock on what's written in this and there's a lot of characters from Sherlock on what's in my life I don't want to say anything that I don't spoil anything. However, it's perfection. Like it's so good I know Kayla just read the first one for her video Which was like reading booktubers famous of the year and I never would have recommended this series to her It's like not her kind of thing if you like Victorian history fantasy kind of books if you like stalking Jack the Ripper This is better than that times a thousand. It has mystery elements, but the mystery isn't what's driving it I would say it's more so the characters relationships These girls I've always said I would die for them. I stand by it. I would die for them and me and my gals We chicken nuggets together. They are my favorite characters that I have ever read and I refuse I can't emphasize you enough how much I refuse to believe this is the end like I'm just not entertaining it and Not allowing that to happen. So I love this. It was five stars This series is gonna be something I reread and reread and reread and reread I love my favorite part of the book is how Catherine one of our characters is writing it So it's all from her Don't have a perspective, but you do you find little nuggets of her in it and the other girls cut in and like Rebuke or complain about what she's writing and go I didn't happen like that and I just love them That's a collective and their relationships with another and how it builds over the course of this I love the kind of like light tongue-and-cheek way We look at some of our classic characters like Sherlock on Watson like Dracula, I think it's like a really refreshing way to look at them and it's just so much fun It's the best series ever if you haven't started yet, please do I love it so much and this is like I've never found a comfort series that I just want to read again and again and again But I think I will definitely reread the first book at some point this year and probably the second one Then I read Super Nova by Marissa Mayer. I gave this truest stars So this is like a superhero versus villain series. This is the last book in the series and II She's not gonna go there. I Did not like the ending to this at all. I didn't feel like it made sense I find the writing in this series to be very basic and not particularly engaging But the ending made zero sense. I don't want to talk about it if you want to hear more about it Here's the vlog that it's in but it was just like the ending was not good It made no sense characters acted in ways that made no sense It wasn't a fitting end to so many of our loose threads in the series So that's what someone has done no matter how much someone has like fucked you over It's fine forgiveness You know what I mean like it just makes no sense. I don't care. I Was angry I was angry It wasn't good. I'm still gonna try out anything that Marissa Mayer writes because I loved the Lunar Chronicles like years ago Maybe I should reread them or reread the first one and if I still like it then I know it's not a problem with her I just had a bit of a problem with this series. I don't think superhero stuff is for me Everyone said this was like the best in a series such a fitting end We're not gonna talk about it anymore. Okay. I'm ready to go home. I'm knackered And then the last book I read this month was The Hand on the Wall by Maureen Johnson This is our last book in the original Trudy Davis trilogy and in this TV our main character is kind of like this young detective Who's trying to figure out the case of the old Ellingham mystery which happened at this school in the 1930s? But she's also trying to figure out kind of modern-day mysteries and I'm giving this what I give this in the end three stars I think yeah, this is another bit of a disappointing ending It's something a lot of people say when reviewing it and I knew that going into it that people aren't happy with who The baddie is and it's like true like it's not it's not a good combination of the three books This is still a really fun murder mystery series. I've said a thousand times and it is a great palette cleanser I really love the mystery elements of it. I love the setting at this old school that was built in the 1930s Which is super opulent and mysterious and it's all these like hidden passages and stuff So there's a lot of elements. I love about it The ending was a bit disappointing and a bit whack like it was a little bit weird Like I still know I still don't really know how that happened didn't really make sense I think the story was too dragged out across the three books And there was not enough elements to it to drag it out across the three books So there we have it. That is my January wrap up let me know down below how you're reading in January when if it went a bit better than mine and Thank you so much for watching. I will see you very very soon. Bye