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

1. onegirl

2. onegirl

@Reaper534 can u help?

3. terenzreignz

Let's just focus on the integral first, keeping in mind that... $\Large \int\limits_1^\infty x^{-\frac43}= \lim_{b\rightarrow\infty}\int\limits_1^bx^{-\frac43}$

4. onegirl

okay

5. terenzreignz

Well, can you integrate $\Large \int x ^{-\frac43}=\color{red}?$

6. onegirl

yes i get -3/3sqrt(x)

7. terenzreignz

This $\Large \frac{-3}{3\sqrt{x}}$?

8. onegirl

yes!

9. terenzreignz

Or this $\Large \frac{-3}{\sqrt[3]x}$

10. onegirl

the second one

11. terenzreignz

lol... it's cube root, not 3sqrt because $$\Large3\sqrt x$$ means something entirely different, ok?

12. onegirl

Okay

13. terenzreignz

Okay, so, we have... $\Large \int\limits_1^b x^{-\frac43}=\left.\frac{-3}{\sqrt[3]x}\right]_1^b$

14. terenzreignz

Can you evaluate this bit?

15. onegirl

i think

16. terenzreignz

Well then, what do you get?

17. onegirl

hmm i'm working on it

18. onegirl

eh...i'm getting a wrong answer :/

19. terenzreignz

Fundamental theorem of Calculus? $\Large \int\limits_a^b f'(x)dx = \left.f(x)\right]_a^b = f(b)-f(a)$

20. onegirl

ohhhh okay

21. terenzreignz

So... $\Large \int\limits_1^b x^{-\frac43} \ dx=\left.\frac{-3}{\sqrt[3]x}\right]_1^b=\color{red}?$

22. onegirl

so i plug in b and 1 into that then subtract?

23. terenzreignz

Yes... but that's not the end yet, just plug in for now, and tell me what you get :)

24. onegirl

okay so -3/3sqrt(b) - -3/3sqrt(1)

25. terenzreignz

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?

26. onegirl

yes

27. terenzreignz

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) $\Large \color{red}{\lim_{b\rightarrow\infty}}\left(-\frac3{\sqrt[3]b}+3\right)$

28. onegirl

okay

29. terenzreignz

So... evaluating the limit...?

30. onegirl

umm i get 3? :/

31. terenzreignz

That's right!

32. onegirl

ohh okay lol

33. onegirl

34. terenzreignz

Well, you technically have two questions, but since there was an answer, then the integral converges, and it converges to 3 ^.^

35. onegirl

okay thanks

36. terenzreignz

No problem :)