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Determine whether the integral converges or diverges. Find the value of the integral if it converges.
 10 months ago
 10 months ago
Determine whether the integral converges or diverges. Find the value of the integral if it converges.
 10 months ago
 10 months ago

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onegirlBest ResponseYou've already chosen the best response.1
@Reaper534 can u help?
 10 months ago

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

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

onegirlBest ResponseYou've already chosen the best response.1
yes i get 3/3sqrt(x)
 10 months ago

terenzreignzBest ResponseYou've already chosen the best response.1
This \[\Large \frac{3}{3\sqrt{x}}\]?
 10 months ago

terenzreignzBest ResponseYou've already chosen the best response.1
Or this \[\Large \frac{3}{\sqrt[3]x}\]
 10 months ago

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

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

terenzreignzBest ResponseYou've already chosen the best response.1
Can you evaluate this bit?
 10 months ago

terenzreignzBest ResponseYou've already chosen the best response.1
Well then, what do you get?
 10 months ago

onegirlBest ResponseYou've already chosen the best response.1
hmm i'm working on it
 10 months ago

onegirlBest ResponseYou've already chosen the best response.1
eh...i'm getting a wrong answer :/
 10 months ago

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

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

onegirlBest ResponseYou've already chosen the best response.1
so i plug in b and 1 into that then subtract?
 10 months ago

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

onegirlBest ResponseYou've already chosen the best response.1
okay so 3/3sqrt(b)  3/3sqrt(1)
 10 months ago

terenzreignzBest ResponseYou've already chosen the best response.1
I'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?
 10 months ago

terenzreignzBest ResponseYou've already chosen the best response.1
Now, 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) \[\Large \color{red}{\lim_{b\rightarrow\infty}}\left(\frac3{\sqrt[3]b}+3\right) \]
 10 months ago

terenzreignzBest ResponseYou've already chosen the best response.1
So... evaluating the limit...?
 10 months ago

onegirlBest ResponseYou've already chosen the best response.1
so thats the final answer?
 10 months ago

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