Help with volume of solid of revolution/integration by parts question

[student93]

Homework Statement

The problem is attached in this post.

Homework Equations

The problem is attached in this post.

The Attempt at a Solution

I've set up the integral via disk method: π∫(e^√x)^2 dx from 0 to 1

I've done integration by parts by don't know how to integrate the second term of the integration by parts which is: ∫(xe^2√x)/(√x) dx from 0 to 1

Also the answer to the question is (π/2)(e^2+1)

 

[LCKurtz]

Homework Statement

The problem is attached in this post.

Homework Equations

The problem is attached in this post.

The Attempt at a Solution

I've set up the integral via disk method: π∫(e^√x)^2 dx from 0 to 1

I've done integration by parts by don't know how to integrate the second term of the integration by parts which is: ∫(xe^2√x)/(√x) dx from 0 to 1

Also the answer to the question is (π/2)(e^2+1)

That problem obviously has a typo in it somewhere. Your integral is correct but it is not an elementary integral and won't match any of the answers.

 

[student93]

I'm pretty sure there's no typo in the question, when I plug in the integral into wolfram alpha, I get the correct answer which is (π/2)(e^2+1)

 

[PeroK]

How did you do the integration by parts? For the integral:

[tex]\int_0^1 e^{2\sqrt{x}}dx[/tex]

Did you think about a substitution?

 

[LCKurtz]

That problem obviously has a typo in it somewhere. Your integral is correct but it is not an elementary integral and won't match any of the answers.

I'm pretty sure there's no typo in the question, when I plug in the integral into wolfram alpha, I get the correct answer which is (π/2)(e^2+1)

Yep. I looked at it too quickly. Try ##2\sqrt x = \ln u## on it.

 

[student93]

How did you do the integration by parts? For the integral:

[tex]\int_0^1 e^{2\sqrt{x}}dx[/tex]

Did you think about a substitution?

u=e^(2√x) dv=dx
du=(e^(2√x))/(√x) dx v=x

∫e^(2√x)dx = xe^(2√x) - ∫((xe^(2√x))/(√x)

 

[PeroK]

I think parts was the wrong way to go. What about the obvious substitution u = √x? In the original integral.

 

[student93]

I don't think u-substitution works in this case, also the directions specifically ask to solve the question via integration by parts.

Also here's the original integral:

π∫e^(2√x) dx, from 0 to 1

 

[PeroK]

I don't think u-substitution works in this case, also the directions specifically ask to solve the question via integration by parts.

The u-substitution does work. It simplifies things ready for the integration by parts!

 

[LCKurtz]

I don't think u-substitution works in this case, also the directions specifically ask to solve the question via integration by parts.

Also here's the original integral:

π∫e^(2√x) dx, from 0 to 1

Substitution does work. Try one of the substitutions that have been suggested and you will see.

 

[student93]

Could you please show exactly how you would do the u-substitution before doing integration by parts? I want to make sure I understand this concept etc.

 

[PeroK]

You could learn about it here:

http://tutorial.math.lamar.edu/Classes/CalcI/SubstitutionRuleDefinite.aspx

In this case:

[tex]u = √x \ , \ x = u^2 \ , \ dx = 2udu[/tex]
And, you don't even have to change the limits.