Question

Take the integral from 0 to 2 of (x^2-4x+4)^(1/2)dx is:
Answer: 2
Please show steps of how to get the answer.

Answer 1

Have you noticed that \(x^{2} - 4x + 4 = (x-2)^{2}\)?

Answer 2

Holy Cow!!!

x-2 is positive on [0,2]

\(\sqrt{x^{2} - 4x + 4} = \sqrt{(x-2)^{2}} = x-2\)

That's quite a bit easier to integrate.

Answer 3

...agreed. I didn't notice that simplification before:/

Answer 4

I tried simplifying it but got -2 instead of 2. :/

Answer 5

So, (1/2x^2 - 2x), and plugged in it's: -2. Good work emele613! :)

Answer 6

But that's not the right answer. :(

Answer 7

It's positive 2.

Answer 8

@amonoconnor That was a lot of work. Good work having the gumption to drag through it!!

Answer 9

Thank you! That's shat I appreciate seeing, so I try and do it for others:)

Answer 10

*what

Answer 11

Whoops!!! x-2 is strictly NEGATIVE on [0,2]. Must have been standing on my head!

\(\sqrt{(x-2)^{2}} = -(x-2) = 2-x\;for\;x\in[0,2]\) Sorry about that.

Answer 12

Oh, I didn't even catch that! Nice one. :P

Answer 13

Once in a while I see it if someone badgers me enough. The ORIGINAL integrand is strictly positive, so it made no sense to get a negative result after simplification.

Answer 14

Thank you everyone! :) I would've not been able to get that.

Answer 15

Well, @amonoconnor 's "Brute Force" method would not have needed this subtle consideration. Kudos again to @amonoconnor for dragging through all that algebra.