Volume of solid with given base and cross sections

In summary, the base of the solid is a semicircle with a radius of 3, and the cross section perpendicular to the x-axis is a square. To find the cross sectional area, the length of the side of the square can be found by taking the difference between the radius and the x-value. The area of the square is irrelevant in finding the volume of the solid, which is equal to 27*pi/4.
  • #1
elsternj
42
0

Homework Statement


The base is the semicircle y = [itex]\sqrt{9-x2}[/itex](Square root of 9-x2.. i don't know why the formatting isn't showing up) where -3 <= x <= 3. The cross section perpendicular to the x-axis are squares.



Homework Equations



-3[itex]\int[/itex]3 = A(y)dy
A(y) = area of cross section

The Attempt at a Solution


Okay so i know the cross section is a square. And i see that y = [itex]\sqrt{9-x2}[/itex] creates a semi-circle. I am not sure how to find the cross sectional area. I'm assuming it needs to be in terms of y. And the area of a square is a (side)2. But how do i know what the side is equal to?
 
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  • #2
Looks to me to be a fairly simply solid (half a cylinder) so I think you're going about it the hard way.
 
  • #3
hmm well the reason I am going about it this way is because the problem is in the section of the book where we are finding volumes by integrating the area of the cross section. I am not aware of another way to do this problem.
 
  • #4
I'm not a big fan of simple problems requiring the use of more advanced math than is needed but I do understand that the math you're studying needs practice problems so I guess you'll need to do it the hard way.
 
  • #5
elsternj said:

Homework Statement


The base is the semicircle y = [itex]\sqrt{9-x2}[/itex](Square root of 9-x2.. i don't know why the formatting isn't showing up) where -3 <= x <= 3. The cross section perpendicular to the x-axis are squares.

Homework Equations



-3[itex]\int[/itex]3 = A(y)dy
A(y) = area of cross section

The Attempt at a Solution


Okay so i know the cross section is a square. And i see that y = [itex]\sqrt{9-x2}[/itex] creates a semi-circle. I am not sure how to find the cross sectional area. I'm assuming it needs to be in terms of y. And the area of a square is a (side)2. But how do i know what the side is equal to?
You are told that the base of the square is in the semi-circle from y= 0 up to [itex]y= \sqrt{9- x^2}[/itex]. That is the length of the side. The area of the square is [itex]9- x^2[/itex].
 
  • #6
but of course when you do it the easy way, the area of the square is irrelevant and the answer pretty much just pops out. When you do it the hard way you should get that the volumn is 27*pi
 

Related to Volume of solid with given base and cross sections

What is the formula for finding the volume of a solid with a given base and cross sections?

The formula for finding the volume of a solid with a given base and cross sections is V = A(x)dx where A(x) represents the area of the cross section at a given x-value and dx represents an infinitesimal change in the x-direction.

How do you find the area of a cross section?

The area of a cross section can be found by using the appropriate formula for the shape of the cross section. For example, if the cross section is a circle, the area can be found using the formula A = πr^2, where r is the radius of the circle.

What is the difference between a solid with a given base and cross sections and a solid with a given base and height?

A solid with a given base and cross sections has varying cross section shapes throughout the solid, while a solid with a given base and height has a constant cross section shape throughout the solid. This means that the volume of a solid with given base and cross sections must be calculated using integration, while the volume of a solid with given base and height can be calculated using the formula V = Bh, where B is the area of the base and h is the height of the solid.

Can the volume of a solid with given base and cross sections be negative?

No, the volume of any solid cannot be negative. Volume is a measure of the amount of space occupied by an object, and since space cannot have negative values, the volume cannot be negative either.

What is the importance of understanding the volume of a solid with given base and cross sections?

Understanding the volume of a solid with given base and cross sections is important in many fields, such as architecture, engineering, and physics. It allows us to calculate the amount of material needed to construct a given solid and to understand how different cross section shapes can affect the overall volume of the solid.

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