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 one year ago
Determine whether the integral converges or diverges. Find the value of the integral if it converges.
 one year ago
Determine whether the integral converges or diverges. Find the value of the integral if it converges.

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onegirl
 one year ago
Best ResponseYou've already chosen the best response.1@Reaper534 can u help?

terenzreignz
 one year ago
Best ResponseYou've already chosen the best response.1Let's just focus on the integral first, keeping in mind that... \[\Large \int\limits_1^\infty x^{\frac43}= \lim_{b\rightarrow\infty}\int\limits_1^bx^{\frac43}\]

terenzreignz
 one year ago
Best ResponseYou've already chosen the best response.1Well, can you integrate \[\Large \int x ^{\frac43}=\color{red}?\]

terenzreignz
 one year ago
Best ResponseYou've already chosen the best response.1This \[\Large \frac{3}{3\sqrt{x}}\]?

terenzreignz
 one year ago
Best ResponseYou've already chosen the best response.1Or this \[\Large \frac{3}{\sqrt[3]x}\]

terenzreignz
 one year ago
Best ResponseYou've already chosen the best response.1lol... it's cube root, not 3sqrt because \(\Large3\sqrt x\) means something entirely different, ok?

terenzreignz
 one year ago
Best ResponseYou've already chosen the best response.1Okay, so, we have... \[\Large \int\limits_1^b x^{\frac43}=\left.\frac{3}{\sqrt[3]x}\right]_1^b\]

terenzreignz
 one year ago
Best ResponseYou've already chosen the best response.1Can you evaluate this bit?

terenzreignz
 one year ago
Best ResponseYou've already chosen the best response.1Well then, what do you get?

onegirl
 one year ago
Best ResponseYou've already chosen the best response.1eh...i'm getting a wrong answer :/

terenzreignz
 one year ago
Best ResponseYou've already chosen the best response.1Fundamental theorem of Calculus? \[\Large \int\limits_a^b f'(x)dx = \left.f(x)\right]_a^b = f(b)f(a)\]

terenzreignz
 one year ago
Best ResponseYou've already chosen the best response.1So... \[\Large \int\limits_1^b x^{\frac43} \ dx=\left.\frac{3}{\sqrt[3]x}\right]_1^b=\color{red}?\]

onegirl
 one year ago
Best ResponseYou've already chosen the best response.1so i plug in b and 1 into that then subtract?

terenzreignz
 one year ago
Best ResponseYou've already chosen the best response.1Yes... but that's not the end yet, just plug in for now, and tell me what you get :)

onegirl
 one year ago
Best ResponseYou've already chosen the best response.1okay so 3/3sqrt(b)  3/3sqrt(1)

terenzreignz
 one year ago
Best ResponseYou've already chosen the best response.1I'm assuming by 3sqrt you mean cube root :D Okay, that being the case, you're right :) \[\Large \int\limits_1^b x^{\frac43}=\frac{3}{\sqrt[3]b}+\frac3{\sqrt[3]1}= \frac3{\sqrt[3]b}+3\] Catch me so far?

terenzreignz
 one year ago
Best ResponseYou've already chosen the best response.1Now, we're supposed to take the improper integral to infinity, right? Remember this... \[\Large \int\limits_1^\infty x^{\frac43} \ dx = \color{red}{\lim_{b\rightarrow\infty}}\int\limits_1^bx^{\frac43} \ dx\] Now is the time to apply that limit (which we haven't done yet) \[\Large \color{red}{\lim_{b\rightarrow\infty}}\left(\frac3{\sqrt[3]b}+3\right) \]

terenzreignz
 one year ago
Best ResponseYou've already chosen the best response.1So... evaluating the limit...?

onegirl
 one year ago
Best ResponseYou've already chosen the best response.1so thats the final answer?

terenzreignz
 one year ago
Best ResponseYou've already chosen the best response.1Well, you technically have two questions, but since there was an answer, then the integral converges, and it converges to 3 ^.^
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