Finding the surface area of a curved object using calculus

[lch7]

1. Homework Statement
I need some help with a surface area of a solid. The solid is made from rotating the line y=x^2 around the x axis. So it's sort of like a cone or a horn. Here are my steps:

2. Homework Equations
Surface of revolution formula
Integrate 2∏r times the square root of 1 plus the derivative squared (dx).

3. The Attempt at a Solution
2$\pi$ $\int$ x$^{2}$ $\sqrt{1+2x^2}$
This is the surface of revolution concept of course. How do I integrate this? Should I make the square root a power of .5??

 

[SteamKing]

If your derivative = 2x, then the derivative squared = (2x)^2, NOT 2x^2

As to the integral itself, try u-substitution with integration by parts.

 

[lch7]

Thanks!

 

[Abhinav R]

As for the curve y = x$^{2}$ , it is an upward parabola with the centre on the origin and x-axis.So when rotated about the x-axis the solid should look something like the attachement image I did.

Now you can integrate the figure using area under curves method.

 

[lch7]

As for the curve y = x$^{2}$ , it is an upward parabola with the centre on the origin and x-axis.So when rotated about the x-axis the solid should look something like the attachement image I did.

Now you can integrate the figure using area under curves method.

No that's not correct sorry

 

[Abhinav R]

No that's not correct sorry

But I think the parabola statement was right,because y = x^2 is an upward parabola right?

 

[lch7]

But I think the parabola statement was right,because y = x^2 is an upward parabola right?

The parabola's base or curve is at the origin, the lines point up left and right. I'm focusing on the parabola's part that is to the right of the y axis. So half of a curve rotated around the origin looks like a curvy cone.

Thanks for you guys' help, I now have the answers. Thanks!

 

[Abhinav R]

The parabola's base or curve is at the origin, the lines point up left and right. I'm focusing on the parabola's part that is to the right of the y axis. So half of a curve rotated around the origin looks like a curvy cone.

Thanks for you guys' help, I now have the answers. Thanks!

Great!