Volume of the solid of revolution

[calchelp]

i need someone to explain to me where i am making a mistake because i am getting an answer that differs from that of the book.


the solid lies between planes perpendicular to the x-axis at x = 0 and x = 4. the cross-sections perpendicular to the axis on the interval 0 ≤ x ≤ 4 are squares whose diagonals run from the parabola y = - sqrt x to the parabola y = sqrt x.


ok so...

the base of the square is a.
the diagonal is 2a².
the diagonal is 2 sqrt x.
so a², the area of the square, is sqrt x.
therefore,
{integral} from 0 to 4 of sqrt x
antidifferentiate and you get 2/3 x^3/2
plug in 4 and 0

final answer: 16/3
book says: 16

where did i mess up?

 

[Subdot]

What is 4^(3/2)? Find that then multiply it by 2. Divide by 3 (because you're multiplying 4^(3/2) by (2/3)) and you'll see what happened.

P.S.: This is not a solid of revolution. Just so you know.

 

[calchelp]

4^(3/2)= 8
8*2=16
16/3= 16/3
i still did not get 16...

 

[Subdot]

Oh, I'm sorry. I read that wrong and flipped the two in my mind. I'll need to think on it some more.

 

[Subdot]

Okay, I have it now. Check the length you have for the diagonal of the square because it is incorrect.

 

[calchelp]

diagonals run from the parabola y = - sqrt x to the parabola y = sqrt x.

the diagonal is the distance from y = - sqrt x to y = sqrt x
how is that not 2 sqrt x?
i honestly can't figure it out.
is there a formula to figure it out?

 

[Subdot]

Yes, the diagonal in terms of x is 2 sqrt(x). However, the diagonal in terms of a you have right now is 2a^2. How did you arrive at this? I presume by the Pythagorean Theorem. However, that does not give you the length of the diagonal. It merely tells you that the sums of the squares of the sides gives you the square of the diagonal (a^2 + a^2 = 2a^2).

What would you need to do to 2a^2 then to get the length not square of the length of the diagonal in terms of a?

 

[calchelp]

i can't believe i made that stupid of a mistake.
so a^2 = 2x
so the volume is 16
thank you so much.

 

[Subdot]

You're welcome! Anytime!