## Bonrozzy Group Title Find the value of k that the line x=k divides the area of the first quadrant region of y=e^-x and the x-axis x >= 1 into two equal parts. 5 months ago 5 months ago

1. myininaya Group Title

So you have talked about improper integrals?

2. Bonrozzy Group Title

yes

3. myininaya Group Title

so we want this: $1/2 \int\limits_{1}^{k}e^{-x} dx=\lim_{a \rightarrow \infty}\int\limits_{1}^{a}e^{-x} dx$

4. myininaya Group Title

we want that one area to be half of that other area

5. myininaya Group Title

did you get this far?

6. Bonrozzy Group Title

i see

7. Bonrozzy Group Title

8. myininaya Group Title

so we can try to evaluate both integrals first then we just solve the equation for k

9. Bonrozzy Group Title

That's true haha it seems so simple now thank you :)

10. Bonrozzy Group Title

I get that k=1, but i know that's not the answer.

11. myininaya Group Title

what 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

12. Bonrozzy Group Title

$-1/e^k = 1/e$

13. Bonrozzy Group Title

sorry it wouldnt be one, but this is the end result i believe

14. myininaya Group Title

that negative is weird killing me because e^(to a positive number) can't be negative

15. myininaya Group Title

i was agreeing it isn't 1 because it would give us 0 for that one area

16. Bonrozzy Group Title

the antiderivative of e^-x is -e^x right?

17. Bonrozzy Group Title

-E^-x*

18. myininaya Group Title

maybe 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

19. myininaya Group Title

yep you are right

20. Bonrozzy Group Title

Okay, 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

21. myininaya Group Title

i got it

22. myininaya Group Title

i wrote the equation just a little wrong

23. Bonrozzy Group Title

oh

24. myininaya Group Title

$\int\limits\limits_{1}^{k}e^{-x} dx=1/2 \lim_{a \rightarrow \infty}\int\limits\limits_{1}^{a}e^{-x} dx$

25. myininaya Group Title

you will get the answer you got using your calculator except it will be exact

26. myininaya Group Title

i'm idiot sorry

27. Bonrozzy Group Title

oh 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!

28. myininaya Group Title

Did you get it?

29. Bonrozzy Group Title

$-1/e^k+1/e=1/2e$

30. Bonrozzy Group Title

thats the function right?

31. myininaya Group Title

yep good so far

32. Bonrozzy Group Title

Then just solve for k?

33. myininaya Group Title

yep! :) I left it as $e^{-k}+e^{-1}=e^{-1}/2$

34. myininaya Group Title

oops with a negative in front of the e^(-k)

35. myininaya Group Title

$-e^{-k}+e^{-1}=1/2e^{-1} => -e^{-k}=-1/2 e^{-1}$

36. myininaya Group Title

get rid of those negatives then take natural log of both sides

37. Bonrozzy Group Title

Yes I got it! Thank you so much :) it's gonna be: $-\ln (1/2e)$

38. myininaya Group Title

or you could write it as ln(2e)

39. Bonrozzy Group Title

true haha

40. myininaya Group Title

but 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

41. Bonrozzy Group Title

Yes I noticed that haha. I can't thank you enough.

42. myininaya Group Title

I'm going to need to learn that cool calculator trick for myself

43. Bonrozzy Group Title

I'm rather inept at using the calculator, but that's one trick I know ;)

44. myininaya Group Title

i know the basics and i know there are really cool tricks in stuff i could learn just never got around to it

45. myininaya Group Title

anyways fun problem we are going over improper integrals today in class :)

46. Bonrozzy Group Title

cool.

47. Bonrozzy Group Title

Well, where i am it's just about dinner time. Thanks again it really saved a lot of hair pulling. See ya