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 11 months ago
Find the value of k that the line x=k divides the area of the first quadrant region of y=e^x and the xaxis x >= 1 into two equal parts.
 11 months ago
Find the value of k that the line x=k divides the area of the first quadrant region of y=e^x and the xaxis x >= 1 into two equal parts.

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myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1So you have talked about improper integrals?

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1so we want this: \[1/2 \int\limits_{1}^{k}e^{x} dx=\lim_{a \rightarrow \infty}\int\limits_{1}^{a}e^{x} dx\]

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1we want that one area to be half of that other area

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1did you get this far?

Bonrozzy
 11 months ago
Best ResponseYou've already chosen the best response.1I hadn't thought about constructing it that way. It makes sense.

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1so we can try to evaluate both integrals first then we just solve the equation for k

Bonrozzy
 11 months ago
Best ResponseYou've already chosen the best response.1That's true haha it seems so simple now thank you :)

Bonrozzy
 11 months ago
Best ResponseYou've already chosen the best response.1I get that k=1, but i know that's not the answer.

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1what is the answer? also yep if k is one, then that one area is 0 and we know 0 is not half the other area

Bonrozzy
 11 months ago
Best ResponseYou've already chosen the best response.1sorry it wouldnt be one, but this is the end result i believe

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1that negative is weird killing me because e^(to a positive number) can't be negative

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1i was agreeing it isn't 1 because it would give us 0 for that one area

Bonrozzy
 11 months ago
Best ResponseYou've already chosen the best response.1the antiderivative of e^x is e^x right?

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1maybe since we can't solve that equation for k there is no way possible to divide the area with a vertical line so we have two equal areas on both sides

Bonrozzy
 11 months ago
Best ResponseYou've already chosen the best response.1Okay, well here's some help. plugging the e^x in my calculator and then taking the integral of that I find the area, i divided it by 2 and then checked the x y table to find the x that gives half the area. It is about 1.693

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1i wrote the equation just a little wrong

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1\[ \int\limits\limits_{1}^{k}e^{x} dx=1/2 \lim_{a \rightarrow \infty}\int\limits\limits_{1}^{a}e^{x} dx\]

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1you will get the answer you got using your calculator except it will be exact

Bonrozzy
 11 months ago
Best ResponseYou've already chosen the best response.1oh such an easily overseen mistake! Ill check this out right now hahhaaha it's perfectly fine :D thanks for taking the time to figure it out!

Bonrozzy
 11 months ago
Best ResponseYou've already chosen the best response.1thats the function right?

Bonrozzy
 11 months ago
Best ResponseYou've already chosen the best response.1Then just solve for k?

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1yep! :) I left it as \[e^{k}+e^{1}=e^{1}/2 \]

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1oops with a negative in front of the e^(k)

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1\[e^{k}+e^{1}=1/2e^{1} => e^{k}=1/2 e^{1} \]

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1get rid of those negatives then take natural log of both sides

Bonrozzy
 11 months ago
Best ResponseYou've already chosen the best response.1Yes I got it! Thank you so much :) it's gonna be: \[\ln (1/2e)\]

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1or you could write it as ln(2e)

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1but yep if you put that in your calculator you will see that its approximation is the same as the approximation you got using just the calculator to find what was it 1.6 something

Bonrozzy
 11 months ago
Best ResponseYou've already chosen the best response.1Yes I noticed that haha. I can't thank you enough.

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1I'm going to need to learn that cool calculator trick for myself

Bonrozzy
 11 months ago
Best ResponseYou've already chosen the best response.1I'm rather inept at using the calculator, but that's one trick I know ;)

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1i know the basics and i know there are really cool tricks in stuff i could learn just never got around to it

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1anyways fun problem we are going over improper integrals today in class :)

Bonrozzy
 11 months ago
Best ResponseYou've already chosen the best response.1Well, where i am it's just about dinner time. Thanks again it really saved a lot of hair pulling. See ya
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