Can't tell if seriously being dumb or not, but OT said integrate (area under the curve, ahem)
@lollol656: First, we must check if it converges, as x to 1 from the left, it approaches -infinty, and as it approaches from right, it goes to infinity so it converges (we're checking for discontinuities), so at 1, it becomes impossible to integrate... In this state.
Regarding @hyperfire7's advice, you can break it into two and do limits as it approaches 1, or, an easier method in this case would be the substitution method.
So instead of integrating (x-1)^-(2/3) from 0 to 2, substitute with u = x - 1, and it becomes integrate (u)^-(2/3) from u = -1 to 1 (because the limits of integration also are affected by the substitution)
From here, the antiderivative is going to be 3 * u^(1/3), and from -1 to 1, so that would be 3 * (1)^(1/3) - 3 * (-1)^(1/3) = 3 + 3 = 6.
The answer is 6, whether you use this technique or @hyperfire7's technique.
[quote=lollol656]so the anti-derivative is 3(x-1)^(1/3). how do I know if it converged?[/quote] Solve it all the way through. If either integrals give an undefined answer, it diverges. I solved it, the answer should be 6.
[quote=OhShutUp][url=http://www.wolframalpha.com/input/?i=integral+%280%2C2%29+1%2F%28x-1%29%5E%282%2F3%29+dx]it's been like two years since i've been in calc but idk try this[/url] lmao i don't remember a thing sorry if that is no help[/quote] Shouldn't get an imaginary answer b/c the integral converges.
[url=http://www.wolframalpha.com/input/?i=integral+%280%2C2%29+1%2F%28x-1%29%5E%282%2F3%29+dx]it's been like two years since i've been in calc but idk try this[/url] lmao i don't remember a thing sorry if that is no help
Alright so you can't integrate this normally since when x=1, the integration will give an undefined answer. Break it apart using limits, and let's call the function f(x): (lim a->1- of integral f(x) with bounds 0 and a) + (lim b->1+ of integral f(x) with bounds b and 2). Everything else should be stuff from Calc I.
6 Comments • Newest first
[quote=CursedVoyboy]Times x by 2/3 equals 1.
Lol s0 dumb[/quote]
Can't tell if seriously being dumb or not, but OT said integrate (area under the curve, ahem)
@lollol656: First, we must check if it converges, as x to 1 from the left, it approaches -infinty, and as it approaches from right, it goes to infinity so it converges (we're checking for discontinuities), so at 1, it becomes impossible to integrate... In this state.
Regarding @hyperfire7's advice, you can break it into two and do limits as it approaches 1, or, an easier method in this case would be the substitution method.
So instead of integrating (x-1)^-(2/3) from 0 to 2, substitute with u = x - 1,
and it becomes integrate (u)^-(2/3) from u = -1 to 1 (because the limits of integration also are affected by the substitution)
From here, the antiderivative is going to be 3 * u^(1/3), and from -1 to 1, so that would be 3 * (1)^(1/3) - 3 * (-1)^(1/3) = 3 + 3 = 6.
The answer is 6, whether you use this technique or @hyperfire7's technique.
[quote=lollol656]so the anti-derivative is 3(x-1)^(1/3). how do I know if it converged?[/quote]
Solve it all the way through. If either integrals give an undefined answer, it diverges.
I solved it, the answer should be 6.
so the anti-derivative is 3(x-1)^(1/3). how do I know if it converged?
[quote=OhShutUp][url=http://www.wolframalpha.com/input/?i=integral+%280%2C2%29+1%2F%28x-1%29%5E%282%2F3%29+dx]it's been like two years since i've been in calc but idk try this[/url]
lmao i don't remember a thing sorry if that is no help[/quote]
Shouldn't get an imaginary answer b/c the integral converges.
[url=http://www.wolframalpha.com/input/?i=integral+%280%2C2%29+1%2F%28x-1%29%5E%282%2F3%29+dx]it's been like two years since i've been in calc but idk try this[/url]
lmao i don't remember a thing sorry if that is no help
Alright so you can't integrate this normally since when x=1, the integration will give an undefined answer.
Break it apart using limits, and let's call the function f(x):
(lim a->1- of integral f(x) with bounds 0 and a) + (lim b->1+ of integral f(x) with bounds b and 2).
Everything else should be stuff from Calc I.