[12 pts] Evaluate the limits Hint: write them as definite integrals and then evaluate the integrals.lim n+02 +1) 3 2 + lim n-+00 2 V1+ 3ilim n-+00 Vn2 +32 i=1...

Write the limit as a definite integral on the interval $[a, b],$ where $c_{i}$ is any point in the ith subinterval.
Limit $\qquad$ Interval
$\lim _{\|\Delta\| \rightarrow 0} \sum_{i=1}^{n} \sqrt{c_{i}^{2}+4} \Delta x_{i} \quad$ $[0,3]$

In problem level. We have to write the ransom is a definite integral and the human in terror that is the given some is unique. None of the partition at 40 screen zero is the mission I've from one of the end. See uh Spirit nice war whole scared James Delta X. I. As a definite integral on the internal zero A group three. But this we are going to use the definition. Uh huh. Are definitely to them there is by definition of a definitely integral air up to be F. F. S. D. S. Is equal to limit. No uh the partition approaching school. Do you know if uh see uh giants delta. It's so let's call it the question one now comparing The sun in equation one with the U. N. Sun we see that F. C. I corresponds food see I. Sphere less for old spirit. Mm So we can write we S. F. C. And is equal to stent. See I severe nice. Uh huh. Uh became also you written S there you go. X. Is equal. True. That's good. Less flour four security. No it was also given that the internal of intergenerational up to be this equal to zero of pills. Mhm. So they didn't some In the three canals unique. No of the partition approaches to zero submission I from one of the and see a spare. Yes. Yeah or scary James Delta. It's then we have done in the form of a definite into their repairs Integral of Integration gave me zero up to three. So we did getting trans zero of the three and the function is explored nice. Uh whole scary B. X. This is very quiet, isn't it?

So we're looking at the limit as X approaches to of this function knows, I rewrote the cube root, um, as to the 1/3 power. So as I plug in to and for both thes that bottom is definitely, you know, to minus two equals zero, and then the top, Well, three times to six plus two is eight the two. The 1/3 power is too young going right. That, um no, the cube a debate is to two months to is zero. So that's our indeterminant form 0/0, which means we can take the derivative of the top. You actually see why I wrote in this way? Because now we have the chain RL 1/2 Sorry. 1/3 leave what's inside alone when you subtract one from 1/3 to get the native 2/3 power, the derivative of Native I forgot to do the changeable derivative of the inside his times. Three. Can't you see those two pieces cancel up the derivative of negative to zero and then derivative of the bottom is one. So now when we plug in bringing, actually, I mean, this is still going to be eight. Um, but Now it's a cube root of eight, uh, to the native second power. Um, what's just like before? Like you motivate us to, um the native power makes it a fraction, and two squared is for so 1/4 is our final answer. And hopefully didn't skip enough step. I showed it off work that this made sense.

We have question number three invades. We need to evaluate uh from limit 2 to 3. Access choir dx has limits of some Okay limit of Sam's So we'll be using on the property or formula to be affects equal to limit. Sorry, b minus a first. B minus a limit and approaches to infinity one by N F A place. F A plus H. Up to have a plus and minus one. H. Being fair at equal to b minus a by and no in this case equal to two, be equal to three. So it will be equal to three miles to buy N that we went by and so let's get started over here and and affects is of course access squad. So b minus a is three minus two. That is one of only two. Right limit and approaches to infinity one by N F a that F two which is four plus F a play set. So two plus one by n whole square bless on this april set enable us to it to place to buy N whole square up to two plus n minus one to place and the minus when one by and that is animals. One by N whole square. Okay, So this will be limited. Any protest to infinity one by N. There's a two square plus two plus one by any whole square that we have already done. So this should be equal to to a Squire. Bless to a square Place. One. Buy and hold square. Uh Just if you open the bracket one buy and hold square place to into two and 21 by N. Similarly last will be uh to a Squire place and mine is one by N. Holds quiet plus two and +22 and minus one by N. Okay, so this will be a limit and approaches to infinity one by N. We should right to a Squire. Plus to a Squire. Plus to a Squire. Up to in terms up to in terms and then Okay, bless one by n whole square plus to buy and hold squared. Bless. And the minus one by n. Hold square. Yeah plus two into two. One by end. Place to buy end. Place three by N players and in two and minus one by and Okay, so this is a limit and approaches to infinity one by N. This will be two X squared plus two X squared plus two, esquire so four and plus one by esquire one square blessed to Squire to a squared plus three square place in the minus one whole square. Okay? And minus one. Hold squared plus four by N. One plus two plus three up to and minus one. So, this will be a limit. N approaches infinity. This is one by N four and plus one minus one by N in two to minus one by N two miles one by end. And uh and should be here one by N Foreign plus and into So divided by six bless four in minus four by two. So this will be limit and approaches to infinity one by N four plus one by six, one of minus one by end, two minus one by N three minus one by N plus two minus two by N. So this will be simply four plus. There's a two by six, two by six plus two. That is 19 by three. Should be there. Answer. Thank you so much.

Let's let's F. Of X. Equal to X squared lasts two X -3. Be a continuous function Virginia's frank Shane on the interval -1- two. Then a petition P from X. Notes X. 12 extend which grace uh based up integral From -1- two. You too. And uh youtube gloves. So I intervals we have Eggs Nuts X. one Which is equal to negative one X. One. Then X. one x 2 two eight N -18 Spain. Which gives size X. N. My last one mm. So of length of length. So the length we have This which is equal to X. one -X. Notes. This is equal to X. one Plus 1. Then Length two will be equals Eggs to my legs. Eggs one. So they ain't the same. This will be equal to eggs and mine. It's eight and 9. It's one which is to minus extend my next one. So from from previous resorts it implies that The integral from negative 1 to 2 of F. Of eggs. The eggs it's equal to the limits. So you have your petition norm of your petition approaches zero. You have is the with P. And this is equal to the eliminates. That's the number of P approaches zero. You have. Yes. So we have X. To one. Your length x. one. Let's F. Start soon. And lane eggs too. So they ain't them with P. So here p the way we define what P. Is. Where were the norm of P. It's equal to much of your length and X. To. Okay the interval for that is from X. I. Thanks I -1 two X. A. So for I equal to 1 2 to A. So then this day implies that thursday implies that there limits applying to our limits. So we have the limits S. P. The norm of P approaches zero. You have your fine Shane. That's his ex one square Plus two x 1 minus three engine X. I. Platz We have Eggs two Squared less. Two eggs two nine is three. Clean eggs too. Last two. They ain't the same. So you have eggs and squared Last two x. m. My last three. This will be equal to dancing girl from negative one two. Soon you have X squared Plus two X -3. The eggs as our results.