1. ## Convexity and integrals

I need help in three basic problems. (But unfortunatelly I can't solve them!)

1. Where is the following function convex/concave? +infection points
$y=\frac{1}{\ln(x)}$

2.What is the integral of the following function?
$f(x)=\frac{\sqrt{1+x^2}}{x}$

3.Determine the surface of revolutionif we rotate the function around the x-axis!
$y=e^{-x}$, $\left[0,\infty\right[$

Thank you very much in advance!
Happy new year!

2. Originally Posted by doug
1. Where is the following function convex/concave? +infection points
$y=\frac{1}{\ln(x)}$
Study the sign of $y''$.

Fernando Revilla

3. Originally Posted by doug
3.Determine the surface of revolutionif we rotate the function around the x-axis!
$y=e^{-x}$, $\left[0,\infty\right[$
Find $V=\pi\int_0^{+\infty}e^{-2x}\;dx$

Fernando Revilla

4. Originally Posted by doug
2.What is the integral of the following function?
$f(x)=\frac{\sqrt{1+x^2}}{x}$
Use the substitution $x=\tan t$ .

Fernando Revilla

5. Originally Posted by doug
I need help in three basic problems. (But unfortunately I can't solve them!) ...

2.What is the integral of the following function?
$f(x)=\frac{\sqrt{1+x^2}}{x}$
...
Thank you very much in advance!
Happy new year!

I found that for me, the substitution $\displaystyle u=\sqrt{1+x^2}$ works well.

In my opinion, the trig substitution, $\displaystyle x=\tan(t)\,,$ gives a result which is still rather difficult to work with.

6. Originally Posted by doug
I need help in three basic problems. (But unfortunately I can't solve them!) ...

3.Determine the surface of revolution if we rotate the function around the x-axis!
$y=e^{-x}$, $\left[0,\infty\right[$

Thank you very much in advance!
Happy new year!

I see that FernandoRevilla gave the formula for volume, not surface area.

Surface Area, S, for a solid of revolution produced by revolving a suitable function, f(x), about the x-axis is: $\displaystyle S=2\pi\int_a^b{f(x)\sqrt{1+\left(f'(x)\right)^2}}\ ,dx\,.$

For, $\displaystyle f(x)=e^{-x}\,,\ \ S=2\pi\int_0^{\infty}{e^{-x}\sqrt{1+e^{-2x}}}\,dx\,.$

7. Originally Posted by SammyS
I see that FernandoRevilla gave the formula for volume, not surface area.
Right (I misread the question). Thanks.

Fernando Revilla