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AP Calc 4.3b: Riemann Sums More Examples

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hello everybody, chapter 4.2 you watch the video 4 Hour going to Let's Get It Started vegan evaluating as I said before more intuitive I'll be one more example of this wiki guide hey we are at 2 evaluate the integral Leominster enter in a girl is -2 to 1 identify a few different things no problem are Delta X find Delta X by doing a B minus a n is 1 - 2 - 2 / end which ends up getting at 3:10 and I see some pie are book refers to the nasty so by we can find that by doing upper bound and lower bound drak X I feel and everything for this problem the lower Bound 2 mini paper Delta app and I'm Batman v of -2 + 3i / n going from the integral to right now we're going to we're going to go from there. How we do that this stuff is just rewriting the integral battered women I'm here can we do that paper up a little bit remind me in a real atom when it looks like this are we take the in a room predestination I equals one of our function at the Feast of I value turned black remember FRP I well we have to look at what original at the next was original a Quebec is going to be to access the original function we are given in the inside dang change to FM time the TI and remember CI is this right here that we found earlier what is equal to the Limit as n approaches Infinity information and equals 1 upper GI is really to time because I with is -2 + 3i / n Pendleton Act i n Delta X and our case and from this point on Meridian properties completely simplified from here have this whole piece it's not going away and it takes a long time to say to write it that party over here however going to get in -4 my first distributor to write 12 / n and now next we're going to take this 3 over we're going to distributor not again -2 / n 1209 * 3 with me 36i and then end times I will get me an squared what I can do here to times 3 I have to get me a stick I heard of me I am out of here when I do three times to get 18 I know where I can see Road Eastern in here has a 6 / n but I can factor out so I can rewrite thing and I'm going to factor out of this nation 6 / n and when I do that I'm going to be left with -2 A3 I over at the bottom now I'm kind of interested I can take out my Constance and I can get it for maintenance so I'm going to be left with everything is on the outside right now it's still there but now I'm going to take I'm going to take - 2 on the outside I'm what I am going to do that because that's going to be really easy I'm going to do decimation of 3 / n outside take out the costumes that are helpful there and know what I'm going to do is and I'll show you what I mean apple pie just keep right here the summation of 1 remembering summation of 1 ingest an org recall that you also have another property can you hear me alright is that n outside + + 1/2 I'm going to run in here instead of what's a + 3 / n outside all of this stuff and outside n + 1 / 2 okay now we're getting somewhere are we still have outside pieces didn't go anywhere and I still have a negative stew and now things in here kennametal MasterCard 3 m squared + 3n over and what I need to do from there take my six over and out pride and I when I do that I'm going to get -12 and over and write 6 and -12 in and then I'm going to get sick time 3 which would be 18 and squared and then 6 x 3 and would be 18 n and a half N squared and a game that comes from I know what are you doing is right here with my n on top I'm here I can write this out as two separate fractions I can write out and 18 and squared over 2 + squared + 18 + 10 10 squared I know this is kind of easier to work with I can see you but what I really have going on Umpqua River and prayers goes away and he can get me 9 I'm here and I'm clocked out but I still have an ad on the bottom and 1809 do I have a plus I'm 9 over and here I still have a negative 12 and I'm still taking a limit and that's the last thing I get to do now when I look at this I take the limit as n approaches infinity and remember I'm going to watch Wisconsin. -12 + 9 is -3 and that is my answer as my evaluation of an integral let's talk about areas of common geometric figures again thank you Tina support it doesn't hurt to see things more than one and we are going to start with the integral from -1 to 2 at 7 and then we're going to take the integral from 2 to 5 of 3 x -6 Super Heroes 16 minus x squared waggle 16 minus x squared what would happen if I graft if I graph the function y equals 7 then I would be looking at 1 2 3 4 5 6 7 8 horizontal line at the Y value 7 Define appear and then I'm looking at the X values from -1 or call that guy there Airline O'Hare and I can look at the phone I can tell that it 33 to the negative 1 to 2 3 and to go from 0 to 7:03 in a length of 7 find the area of a rectangle are multiplied the whip area equals length 3 times 7 equals 21 this year I need to grab this line 3x - 6 and I can do that going to make kind of a really long way I am going to start at -6 John 1 2 3 4 5 6 make a point 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 call that 112 I'm down here at -6 and I have a slope of 3 went to go up one two three one two three one two three everyone looks like my x intercept is happening at 212-3131 1 2 3 1 I don't know I'm good together once I connect those point I can take a look at my boundaries I'm interested from to download a climber my boundaries are that last point that we made and so I can see in that area I'm looking for a triangle right here I can recall that the area of a triangle is 1/2 time to hike our kids that's 1/2 times and days which is 3 is everyone up 3 for everyone the appointment we went up by 3 6 9 and what we can tell from here then is that we go get 27 / 2 a princess and a girl we're going to be doing for this interval here it's important that you recognize this is a half circle y equals radical we actually have a half circle with the radius or centered at the origin 1 2 3 4 go out one two three four and go up one two three four acres circle is centered around the origin and I go up and over to my other and I stop I'm interested in the area of the tax Circle here remember that the area of a circle is pi r squared but we're not x 1/2 Enterprise buy a radius is 4 for square to 16th do I get 16 x 108 and then fry in my area video additive interval property absolute value function and Beyond interval a property underline that part 2 where am I to do an example of into evaluate the integral -5 to 0 of the absolute value of x + 3 in order to get I want to review some properties it might not be so friend I wouldn't even start you off at alright so I what I want you to understand we have like an f of x equals absolute value x + 1 coordinate plane great it means that I have a Vertex at -1 transformations or leftover money there are in here so I'd go left one that's my Vortec and then I have a cloak of whatever's next to act like the one in front of Acts here so I have one eye that means I'm going to 1/1 of 1/1 of 1/1 and in the other direction I'm going to go up one over one of one 1 / 1 aminophylline / 1 Apple Valley pumpkin puree that classic little triangle thing that it always done is going to have that vertex at 1 color hair this 1/2 - 1 and the can I buy the half of the graph is represented by the function x + 1 artograph for when were looking at the absolute value of x + 3 and I guess I'll do the whole interval x + 3 b x we're going to go and we're going to make that vertex happen at night drink 1 2 3 4 5 6 over there also is -3 hear back from Reno Arbor Texas going to be slope in front of X is going to be a one and four we're going to go up one over one in both directions of 1/1 I'm going to go over one wants more up one over one and I can see you then triangle and over here I'm going to go back with me when you go up one over one that's one of the 1 / 1 and then I have my absolute value function how to split into two pieces I have a triangle here -3 and I have a triangle here I -3 5 right cuz that's where I went shopping with a negative 5 ignoring me but I want you to see this is going on here remember the area of a triangle is just one half BH so the base of the first one over here to the right Zachary is also Street we can find the area to be a 9 times 3 times 9 to the area of triangle over here and what do we have a base of 2 Amy x 1/2 and I'm going to leave it as a fraction for now 2 * 2 is 4 + 4/2 now we need to do is area Tire interval friends that means I have to add the area of this triangle and the area of and I get 13 / 2 I have an interview I want you guys to remember that this side of the function is represented by the function and is represented as negative as minor as an integral looks like really what we can do who is the adverse property from negative 5 negative we're taking the integral of the negative function of negative x - 3 adding that to the integral from -3 to of the positive x + 3 exactly are you found that this area was at 9 or anytime of the area here was 9 / 2 Anthony come to the same conclusion either way one more example again focus on using the integral properties hey we were asked to evaluate the integral from 2 to 4 of X cubed minus 3x 7 Deer using the following value at night from 2 to qdx 6000 XVI in thick and from 2 to 4. who is going to be helpful to realize that this is the same one I'm here so we can use these things to help you can rewrite the integral of x cubed minus free outside the integral 2 to 4 of x the integral of 2 3/4 of 7 and we can evaluate the given information I know we're doing three times the ACT is 63 * 6 is 18 find a used car, you can take out the 7 and take the integral of 1 react let me know the integral of 1 x 2 here at 7:14 when I add all these numbers together I guess and there you have it

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AP Calc 4.3b: Riemann Sums More Examples

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