Volume of the solid of revolution

In summary, the solid in question has a base that 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. The formula for the diagonal in terms of x is 2 sqrt(x). However, the mistake was made in calculating the diagonal in terms of a, which should have been 2a. Once this mistake was corrected, the correct answer of 16 was obtained.
  • #1
calchelp
11
0
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 x3/2
plug in 4 and 0

final answer: 16/3
book says: 16

where did i mess up?
 
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  • #2
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.
 
  • #3
4^(3/2)= 8
8*2=16
16/3= 16/3
i still did not get 16...
 
  • #4
Oh, I'm sorry. I read that wrong and flipped the two in my mind. I'll need to think on it some more.
 
  • #5
Okay, I have it now. Check the length you have for the diagonal of the square because it is incorrect.
 
  • #6
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?
 
  • #7
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?
 
  • #8
i can't believe i made that stupid of a mistake.
so a^2 = 2x
so the volume is 16
thank you so much.
 
  • #9
You're welcome! Anytime!
 

FAQ: Volume of the solid of revolution

What is the "volume of the solid of revolution"?

The volume of the solid of revolution is the measure of the space occupied by a three-dimensional object that is formed by rotating a two-dimensional shape around an axis. This concept is commonly used in calculus to find the volume of objects such as cylinders, cones, and spheres.

How is the volume of the solid of revolution calculated?

The volume of the solid of revolution can be calculated using the disk method or the shell method. The disk method involves dividing the shape into thin horizontal slices, calculating the volume of each slice, and then adding them together. The shell method involves dividing the shape into thin vertical strips, calculating the volume of each strip, and then adding them together.

What is the difference between the disk method and the shell method?

The main difference between the disk method and the shell method is the way the shape is divided. The disk method divides the shape into horizontal slices, while the shell method divides it into vertical strips. Depending on the shape and the axis of rotation, one method may be more convenient to use than the other.

What are some common shapes used to find the volume of the solid of revolution?

Some common shapes used to find the volume of the solid of revolution include circles, rectangles, and semicircles. These shapes can be rotated around different axes to create objects such as cylinders, cones, and spheres. Other more complex shapes can also be used, but their volume calculation may require more advanced techniques.

How is the volume of the solid of revolution used in real-life applications?

The volume of the solid of revolution has many real-life applications, such as in engineering, architecture, and physics. For example, it can be used to calculate the volume of objects such as pipes, tanks, and buildings. In physics, it can be used to find the volume of 3D objects like planets and stars.

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