# Finding the surface area of a curved object using calculus

1. Nov 11, 2012

### lch7

1. The problem statement, all variables and given/known data
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. Relevant 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??

2. Nov 11, 2012

### SteamKing

Staff Emeritus
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.

3. Nov 11, 2012

### lch7

Thanks!

4. Nov 11, 2012

### 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.

#### Attached Files:

• ###### y=x^2 solid.pdf
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5. Nov 11, 2012

### lch7

No that's not correct sorry

6. Nov 11, 2012

### Abhinav R

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

7. Nov 11, 2012

### lch7

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!

8. Nov 11, 2012

Great!!