## anonymous 3 years ago Determine whether the integral converges or diverges. Find the value of the integral if it converges.

1. anonymous

2. anonymous

@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. anonymous

okay

5. terenzreignz

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

6. anonymous

yes i get -3/3sqrt(x)

7. terenzreignz

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

8. anonymous

yes!

9. terenzreignz

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

10. anonymous

the second one

11. terenzreignz

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

12. anonymous

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. anonymous

i think

16. terenzreignz

Well then, what do you get?

17. anonymous

hmm i'm working on it

18. anonymous

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. anonymous

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. anonymous

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. anonymous

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. anonymous

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. anonymous

okay

29. terenzreignz

So... evaluating the limit...?

30. anonymous

umm i get 3? :/

31. terenzreignz

That's right!

32. anonymous

ohh okay lol

33. anonymous

34. terenzreignz

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

35. anonymous

okay thanks

36. terenzreignz

No problem :)