Question

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

Answer 1

What's the integral?

Answer 2

\[\int_1^\infty x^{-1/3}~dx\]
Write as a limit:
\[\lim_{b\to\infty}\int_1^b x^{-1/3}~dx\]
Then, like with the last one, apply the power rule and FTC:
\[\lim_{b\to\infty}\left[\frac{x^{2/3}}{\frac{2}{3}}\right]_1^b\\
\frac{3}{2}\lim_{b\to\infty}\left[x^{2/3}\right]_1^b\\
\frac{3}{2}\left(\lim_{b\to\infty}b^{2/3}-1^{2/3}\right)\\
\]

Answer 3

okay so it converges

Answer 4

okay thanks

Answer 5

No, it diverges. \[\lim_{b\to\infty}b^{2/3}=\infty\]

Answer 6

okay

Answer 7

so that last one was diverges too since the lim didn't exist and it wasn't finite?

Answer 8

Divergent.\[\int\limits_{1}^{\infty}x^{-\frac{1}{3}}dx>\int\limits_{1}^{\infty}x^{-1}dx\]Which is divergent.

Answer 9

okay

Answer 10

got you