Question

Alright I integrate x^4+2/x^4 to get
-2/3x^3 + x^5/5 now how do evaluate it at 0 to 1 and prove if it diverges?

Answer 1

www.gummy-stuff.org/Calculus/calculus-3.pdf- for better explanation

Answer 2

if you integrate it
\[x^{5}/5-2/3x^{3}\]

Answer 3

Is it a definite integral?

Answer 4

[(-2/3x^3 + x^5/5) - (-2/3x^3 + x^5/5)]
Where in the left hand side you plug in 1
and in the right hand side you plug in 0

Answer 5

\[\int\limits_{0}^{1}x^4+2/x^4 dx\]

Answer 6

That's basically what they are asking you to do.

Answer 7

@Romero that is what they are asking me to solve it's basically if it converges or diverges

Answer 8

it's an improper integral:\[\left[\begin{matrix}\frac{x^5}{5} - \frac{2}{3x^3} \end{matrix}\right]^{1}_{a}\]

\[=\frac{1}{5} - \frac{2}{3} - \frac{a^5}{5} + \frac{2}{3a^3}\]

now as a tends to zero we can say that the last term diverges to infinity

thats how i learnt it anyway :)

Answer 9

yup - thats correct

Answer 10

Well when you evaluate the ingetral from 0 to 1 and you get an exact value then it converges but if in this case since you have x in the denomenator then it diverges.

Answer 11

fun fact with improper integrals:

if you find the volume enclosed from 1 to +infinity when the function
y = 1/x
is rotated about the x axis

you'll get it converging to pi

but if you find the surface area of this function from 1 to +infinity

we find it diverges.

so if it a bucket of paint could be built in this shape we wouldn't be able to paint the outside with the paint we could fit inside

Answer 12

Torricelli's trumpet =)