Evaluate the following definite integrals as limit of sums.
$$
\int_{a}^{b} x d x
$$

Answer

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Calculus 2 / BC

NCERT Class 12 Part 2 - Math

Chapter 7

Integrals

Section 8

Fundamental Theorem of Calculus

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Video Transcript

we have question number one in this we have to evaluate the following definite integral as the middle. So we have to evaluate from A to B. X dx as limited some. So we'll be using the formula A to B affects the X equal to b minus a limit. X approaches to sorry, end approaches to infinity. An apology to infinity. Run by N F A. Place F A passage less and so on. F A place and minus one at We're at equals two B minus a by. And so we'll be using this formula. So now if you compare these two compare these two we will be getting equal to equal to A and B equal to B and we have affects equal to X. Okay, so this will be equal to from to be X dx B minus a. Simply B minus a limit and tends to infinity one by N F A F means we have to plug in a in place of X in effect, so this will be simply a place F A plus one. Okay, I have a placenta, a play set, so A plus H plus a place to it up to a place and minus when at. Okay, No, this becomes equal to b minus a limit, intends to infinity limit and approaches to infinity one by N. We'll be writing a place a place a place up to in terms Yeah, up to in terms in terms class at place, do it plus three. Eight plus up to n minus when H. Okay, so this will become b minus a limit and approaches to infinity one by N. This will become eight times in so in a plus add to a straight up two and minus one edge. So we have to use since this is what we say, this is arithmetic progression with first etch and common difference also at. So the sun will be and by two which is equal to and minus one by two and by two into to a so and by to an apple itself at the place last time is this? So we'll be just writing Okay. And H minus H. So this is N by to a place. L Okay now yes, this is b minus a limit and approaches to infinity one by N. Uh This will be at an AT will be cancel out if you take out in common we'll be having a plus and mine is one by two into H in an environment cancel out. So we'll be getting b minus a limit and approaches to infinity uh and approaches to infinity A plus. Yeah and minus one by two. Into action. Okay, B minus a. And a protest in front. So april cell in the minus one but do what at and two And in 20 minutes? One by two it was good. Mhm. Uh huh. Okay, so this is great. Now we have to plug in the value of at at S B minus a by and or it is right here in place of desk, uh we will be having b minus a limit and approaches to infinity a place n minus one by two and two B minus A by n. So this is b minus a limit and approaches to infinity a place or let us divide this by N. We'll be having one by one minus N by 22 B minus a. Now if any approaches to infinity this will a close to zero. So we'll be left with, we might say inside it will be a plus B minus a by two. Okay, so this is a B minus a to a class, b minus a by two. So B minus a into a plus B by two, B squared minus a squared by two. They should be the correct answer. Thanks.

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