onegirl
Determine whether the integral converges or diverges. Find the value of the integral if it converges.
Delete
Share
This Question is Closed
onegirl
Best Response
You've already chosen the best response.
1
onegirl
Best Response
You've already chosen the best response.
1
@Reaper534 can u help?
terenzreignz
Best Response
You've already chosen the best response.
1
Let's just focus on the integral first, keeping in mind that...
\[\Large \int\limits_1^\infty x^{-\frac43}= \lim_{b\rightarrow\infty}\int\limits_1^bx^{-\frac43}\]
onegirl
Best Response
You've already chosen the best response.
1
okay
terenzreignz
Best Response
You've already chosen the best response.
1
Well, can you integrate
\[\Large \int x ^{-\frac43}=\color{red}?\]
onegirl
Best Response
You've already chosen the best response.
1
yes i get -3/3sqrt(x)
terenzreignz
Best Response
You've already chosen the best response.
1
This
\[\Large \frac{-3}{3\sqrt{x}}\]?
onegirl
Best Response
You've already chosen the best response.
1
yes!
terenzreignz
Best Response
You've already chosen the best response.
1
Or this
\[\Large \frac{-3}{\sqrt[3]x}\]
onegirl
Best Response
You've already chosen the best response.
1
the second one
terenzreignz
Best Response
You've already chosen the best response.
1
lol... it's cube root, not 3sqrt
because \(\Large3\sqrt x\) means something entirely different, ok?
onegirl
Best Response
You've already chosen the best response.
1
Okay
terenzreignz
Best Response
You've already chosen the best response.
1
Okay, so, we have...
\[\Large \int\limits_1^b x^{-\frac43}=\left.\frac{-3}{\sqrt[3]x}\right]_1^b\]
terenzreignz
Best Response
You've already chosen the best response.
1
Can you evaluate this bit?
onegirl
Best Response
You've already chosen the best response.
1
i think
terenzreignz
Best Response
You've already chosen the best response.
1
Well then, what do you get?
onegirl
Best Response
You've already chosen the best response.
1
hmm i'm working on it
onegirl
Best Response
You've already chosen the best response.
1
eh...i'm getting a wrong answer :/
terenzreignz
Best Response
You've already chosen the best response.
1
Fundamental theorem of Calculus?
\[\Large \int\limits_a^b f'(x)dx = \left.f(x)\right]_a^b = f(b)-f(a)\]
onegirl
Best Response
You've already chosen the best response.
1
ohhhh okay
terenzreignz
Best Response
You've already chosen the best response.
1
So...
\[\Large \int\limits_1^b x^{-\frac43} \ dx=\left.\frac{-3}{\sqrt[3]x}\right]_1^b=\color{red}?\]
onegirl
Best Response
You've already chosen the best response.
1
so i plug in b and 1 into that then subtract?
terenzreignz
Best Response
You'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 :)
onegirl
Best Response
You've already chosen the best response.
1
okay so -3/3sqrt(b) - -3/3sqrt(1)
terenzreignz
Best Response
You'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?
onegirl
Best Response
You've already chosen the best response.
1
yes
terenzreignz
Best Response
You'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)
\[\Large \color{red}{\lim_{b\rightarrow\infty}}\left(-\frac3{\sqrt[3]b}+3\right) \]
onegirl
Best Response
You've already chosen the best response.
1
okay
terenzreignz
Best Response
You've already chosen the best response.
1
So... evaluating the limit...?
onegirl
Best Response
You've already chosen the best response.
1
umm i get 3? :/
terenzreignz
Best Response
You've already chosen the best response.
1
That's right!
onegirl
Best Response
You've already chosen the best response.
1
ohh okay lol
onegirl
Best Response
You've already chosen the best response.
1
so thats the final answer?
terenzreignz
Best Response
You'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 ^.^
onegirl
Best Response
You've already chosen the best response.
1
okay thanks
terenzreignz
Best Response
You've already chosen the best response.
1
No problem :)