A community for students.
Here's the question you clicked on:
 0 viewing
onegirl
 one year ago
Determine whether the integral converges or diverges. Find the value of the integral if it converges.
onegirl
 one year ago
Determine whether the integral converges or diverges. Find the value of the integral if it converges.

This Question is Closed

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 ^.^
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.