Area and Volume of a Function Question

[mmtrkb]

f(x) = 1/(x^0.6) , 1< or = x < infinity

Find the area between this curve and the x-axis, if it is finite.

Find the volume of the solid obtained by the revolution of this curve about the x-axis, if it is finite

 

[tkhunny]

What's your plan? Two integrals to set up. Much to ponder.

Hint: Fill it it up but can't paint it.

 

[mmtrkb]

f(x) = 1/(x^0.6) , 1< or = x < infinity

Find the area between this curve and the x-axis, if it is finite.

f(x) = 1/(x^0.6) = lim integral 1 to b x^ - .6
b -> infinity

=lim [-(.6/x^1.6)] 1 to b
b -> infinity

= lim [(-.6/b^1.6) - (-.6/1^1.6)]
b -> infinity

I have no Idea what I am doing, is this even close?

 

[lookagain]

f(x) = 1/(x^0.6) , 1< or = x < infinity

Find the area between this curve and the x-axis, if it is finite.

\(\displaystyle > \ >\)f(x) = 1/(x^0.6)\(\displaystyle < \ <\) = \(\displaystyle > \ >\)lim integral 1 to b x^ - .6, [b --> infinity]\(\displaystyle < \ <\)

\(\displaystyle mmtrkb, \ this \ is \ not \ to \ be \ an \ equality \ here.\)

\(\displaystyle \text{The left-hand side shows the function, }\)
\(\displaystyle \text{while the right-hand side shows an integral.}\)

\(\displaystyle f(x) \ = \ \frac{1}{x^{0.6}}\)


\(\displaystyle \lim_{b \to \infty}{\int_1^b\bigg(\frac{1}{x^{0.6}}\bigg) dx\)


\(\displaystyle If \ you're\ familiar \ with \ the \ \frac{1}{x^p}\ form, \ \ then \ you \ might \ know \\)

\(\displaystyle that \ this \ summation \ does\ not \ converge \ for\ p \ \le \ 1.\)

\(\displaystyle \text{In other words, because 0.6 is not greater than 1, this area is not finite.}\)

 

[tkhunny]

f(x) = 1/(x^0.6) , 1< or = x < infinity

Find the area between this curve and the x-axis, if it is finite.

f(x) = 1/(x^0.6) = lim integral 1 to b x^ - .6
b -> infinity

=lim [-(.6/x^1.6)] 1 to b
b -> infinity

= lim [(-.6/b^1.6) - (-.6/1^1.6)]
b -> infinity

I have no Idea what I am doing, is this even close?

You seem to be headed in the right direction. Now, just think on it until it soaks in. What does that "b-> infinity" mean? While 'b' is finite, what does it mean?