Question

Help with integrals

Answer 1

\[\int\limits_{1}^{8} \sqrt{1+ \frac{ (4x^(2/3) -1)^2 }{ 16x^(2/3) }}dx\]

Answer 2

mistake 4x^(2/3) and 16x^(2/3)

Answer 3

\[\int\limits_{1}^{8}\sqrt{1+\frac{ (4x^{2/3} - 1)^{2} }{ 16x^{2/3} }}dx\]

correct?

Answer 4

yes, thanks you

Answer 5

I'm looking for the length of the curve and after finding the derivative of the first function I ended up here stuck.

Answer 6

Yeah, this looks exactly like an arc length problem. And just as a hint, the square root often ends up being something you had prior to taking your derivative yet with the opposite sign. Somewhere Im guessing you had a subtraction of two fractions before this integral. Usually what happens is the simplification is the same thing but with a plus sign. But let's still work it out ^_^

Answer 7

Yeah I did have 2 fractions which I thought it would help combining them and then squaring them to simplify things

Answer 8

\[\int\limits_{1}^{8}\sqrt{1+(x^{1/3}-\frac{ 1 }{ 4x^{1/3} })^{2}}dx\]

Answer 9

from that I tried simplifying to the first one

Answer 10

\[\sqrt{1 + \frac{ (4x^{2/3} - 1)^{2} }{ 16x^{2/3} }} = \sqrt{\frac{ 16x^{2/3} + (4x^{2/3} - 1)^{2} }{ 16x^{2/3} }}\]
\[\sqrt{\frac{ 16x^{2/3} + 16x^{4/3} - 8x^{2/3} + 1 }{ 16x^{2/3} }} = \sqrt{\frac{ 16x^{4/3} + 8x^{2/3} + 1 }{ 16x^{2/3} }}\]
\[\frac{ \sqrt{(4x^{2/3} + 1)^{2}} }{ \sqrt{16x^{2/3}} } = \frac{ 4x^{2/3} + 1 }{ 4x^{1/3} }\]

I bet that's pretty much what you started with just with a minus sign :)
THat happens a lot in arc length problems, so its a result to be aware of if youre stuck, knowing that the simplification may just be the fractions you started with but with the opposite sign.

Answer 11

hmm what happened to the 16x^(2/3) to give off a +8x^(2/3) on the fourth part?

Answer 12

There's a \(16x^{2/3}\) and a \(-8x^{2/3}\) that combined in the numerator after I foiled out the squared term.

Answer 13

ooo wow I didn't noticed that...

Answer 14

That + 1 in the arc length formula just loves to do that :) So just something to try in the future. If you're stuck, take what you started with, in this case \(x^{1/3} - \frac{1}{4x^{1/3}}\) and see if the simplification of your root is the same just with a sign change.

Answer 15

Literally I was in that part and I totally didn't noticed that --- I'm blind. Thanks you

Answer 16

You're welcome ^_^