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onegirl
 2 years ago
Determine whether the integral converges or diverges. Find the value of the integral if it converges.
onegirl
 2 years ago
Determine whether the integral converges or diverges. Find the value of the integral if it converges.

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

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

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

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

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

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

terenzreignz
 2 years 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
 2 years 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
 2 years ago
Best ResponseYou've already chosen the best response.1so i plug in b and 1 into that then subtract?

terenzreignz
 2 years 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
 2 years ago
Best ResponseYou've already chosen the best response.1okay so 3/3sqrt(b)  3/3sqrt(1)

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

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

terenzreignz
 2 years 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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