washers and discs

[skystar]

please help. i'm completely lost when it comes to this chapter.

one of my questions is:

the sold lies between the planes perpendicular to the x-axis at x=0 and x=4. the cross sections perpendicular to the axis on the interval 0 less than or equal to x which is less than or equal to 4 are squares whose diagonals run from the parabola y= square root (x) to y= negative square root (x).

this is how i did it:
i took the integral from 0 to 4.
the equation inside the integral i wrote as just x because i took the area of the square which is just the length times the length so square root x squred is just x.
the answer i got is 8, the correct answer is 16.

what did i do wrong?

 

[mark07]

At the point x between 0 and 4, diagonal of the square cross-section is given by

\(\displaystyle \L \sqrt{x} - (-\sqrt{x}) = 2\sqrt{x}.\)

Then, if the side of that square has length a, you have

\(\displaystyle \L a \sqrt{2} = 2\sqrt{x}\)

and

\(\displaystyle \L a = \sqrt{2x}.\)

So the area of the cross-section is

\(\displaystyle \L a^2 = 2x.\)

Volume is given by

\(\displaystyle \L \int_0^4 2x\, dx = ?\)