Areas of Surfaces of Revolution

[Jarvis88]

Homework Statement

Find the area of the surface generated by revolving the curve y=√x+1, 1≤ x ≤5, about the x-axis.

I'm stuck trying to figure out how I can use substitution...if I am even able. I was trying to rewrite 1 as 4(x+1)/4(x+1) but still can't seem to get the right terms to cancel to allow me to continue further.

Homework Equations

S= ∫ ⁵₁ 2πy (√1+(^dy_dx)²) dx

3. The Attempt at a Solution

 

[Mark44]

Homework Statement

Find the area of the surface generated by revolving the curve y=√x+1, 1≤ x ≤5, about the x-axis.

I'm stuck trying to figure out how I can use substitution...if I am even able. I was trying to rewrite 1 as 4(x+1)/4(x+1) but still can't seem to get the right terms to cancel to allow me to continue further.

Homework Equations

S= ∫ ⁵₁ 2πy (√1+(^dy_dx)²) dx

3. The Attempt at a Solution
View attachment 111983

I am puzzled by what you did in the 3rd line. In the line above that you have dy/dx, but then switched in the 3rd line to dx/dy. Why?

 

[Jarvis88]

I'm sorry, I was doing a problem previous to this one where it was dx/dy. I meant to use dy/dx throughout the problem.

 

[Jarvis88]

If it helps, this is the answer I found on Chegg.com. However, I don't understand how they got from the 1st step to the 2nd step. I've tried it a few different ways and have yet to end up with what they have. I also don't fully trust sites like that as they are not always correct.

 

[LCKurtz]

View attachment 111989

If it helps, this is the answer I found on Chegg.com. However, I don't understand how they got from the 1st step to the 2nd step.

They use $\sqrt a \sqrt b = \sqrt{ab}$.

 

[Jarvis88]

Ok. I'm getting tired so maybe I've made a silly algebra error somewhere, but I'm getting crazy numbers. I must not be doing an algebra manipulation correctly, but I can't seem to figure out where I've gone wrong in my simplification of the square root.

They use $\sqrt a \sqrt b = \sqrt{ab}$.

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[Mark44]

You have an algebra mistake at the end of the 2nd line on the right half of your work.

$2\pi \int_1^5\frac{\sqrt{4(x + 1) + 1}}{2}dx$
You canceled the 4 in the radical with the 2 in the denominator, leaving a factor of 2 in the radical. This is actually two mistakes.
1) $\frac{ab + c}{a} \ne b + c$ That's essentially what you did.
2) $\frac{\sqrt{4}} 2 \ne \sqrt{2}$

BTW, your problem description threw me off:

Find the area of the surface generated by revolving the curve y=√x+1, 1≤ x ≤5, about the x-axis.

From your work, the function apparently is $f(x) = \sqrt{x + 1}$. In linear form, without LaTeX, what you should have written is f(x) = √(x + 1).

 

[Jarvis88]

BTW, your problem description threw me off:

From your work, the function apparently is $f(x) = \sqrt{x + 1}$. In linear form, without LaTeX, what you should have written is f(x) = √(x + 1).

Honestly, I'm pretty new to all this so I'm not familiar with how to use LaTeX properly. In the problem description I was quoting the description verbatim from my assignment. Thank you for the help! I don't know how I made those silly algebra errors...

 

[Mark44]

Honestly, I'm pretty new to all this so I'm not familiar with how to use LaTeX properly.

See our tutorial -- https://www.physicsforums.com/help/latexhelp/.

Everything you did on paper can be done using LaTeX. We prefer that members post their work directly here rather than as pasted images, as it makes it easier for helpers to identify exactly where errors are made. Also, many of the images that get posted are of poor quality and hard to read. Yours were easy to read, though.

 

[Jarvis88]

See our tutorial -- https://www.physicsforums.com/help/latexhelp/.

Everything you did on paper can be done using LaTeX. We prefer that members post their work directly here rather than as pasted images, as it makes it easier for helpers to identify exactly where errors are made. Also, many of the images that get posted are of poor quality and hard to read. Yours were easy to read, though.

Thank you! Will do!