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Main content
Current time:0:00Total duration:3:52
AP.CALC:
LIM‑6 (EU)
,
LIM‑6.A (LO)
,
LIM‑6.A.1 (EK)
,
LIM‑6.A.2 (EK)

Video transcript

what I want to figure out in this video is the area under the curve y equals one over x squared with one as with x equals 1 as our lower boundary and have no upper boundary just keep on going forever and forever as it really is essentially as X approaches infinity so I want to figure out what this entire area is and one way that we can denote that is with an improper definite integral or an improper integral and we would denote it as 1 is our lower boundary but we're just going to keep on going forever as our upper boundary so our upper boundary is infinity and we're taking the integral of 1 over x squared DX and so let me be very clear this right over here is an improper improper improper integral now how do we actually deal with this well by definition this is the same thing as the limit as the limit as n approaches infinity of the integral from 1 to N of 1 over x squared 1 over x squared DX and this is nice because we know how to evaluate this this is just a definite integral where the upper boundary is N and then we know how to take limits we can figure out what the limit is as n approaches infinity so let's figure out if we can actually evaluate this thing so over the second fundamental theorem of calculus so the second part of the fundamental theorem of calculus tells us that this piece right over here so let me write the limit part so this part I'll just rewrite the limit as n approaches infinity of of and we're going to use the second fundamental theorem of calculus we're going to evaluate the antiderivative of X of 1 over x squared or X to the negative 2 so the antiderivative of x to the negative 2 is negative x to the negative 1 so negative x to the negative 1 or negative 1 over X so negative 1 over X is the antiderivative and we're going to evaluate it at N and evaluate it at 1 so this is going to be equal to the limit the limit as n approaches infinity let's see if we evaluate this thing a 10 we get negative 1 over N so we get negative 1 over N and from that we're going to subtract this thing evaluated 1 so it's negative 1 over 1 or it's negative 1 so this right over here is negative 1 and so we're going to find the limit as n approaches infinity of this business this stuff right here is just this stuff right here I haven't found the limit yet so this is going to be equal to the limit as n approaches infinity let's see this is positive 1 positive 1 and we could even write that minus 1 over n of 1 minus 1 over N and lucky for us this limit actually exists limit as n approaches infinity this term right over here is going to get closer and closer and closer to zero 1 over infinity you can essentially view as zero so this right over here is going to be equal to 1 which is pretty neat we have this area that has no right boundary just keeps on going forever but we still have a finite area and the area is actually exactly equal exactly equal to 1 so in this case we had an improper integral and because we were actually able to evaluate it and come up with a number that this limit actually existed we say that this improper integral right over here is convergent convergent if for whatever reason this was unbounded we couldn't come up with some type of a finite number here if the area was infinite we would say that it is divergent so right over here we were figuring we figured out a kind of neat thing this area is exactly 1
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