Volumes Question (1 Viewer)

[echelon4]

I can't seem to get the right answer for this question:

The base of a certain solid is the region bounded by the curve y=x^2 and the line y=x. Cross sections of this solid by planes perpendicular to the x-axis are squares. Find the volume of the solid.

I keep getting the answer as 2/5, but the correct answer seems to be 1/30. Can neone get the right answer? Can you post your working out?

Thanks in advanced

 

[_ShiFTy_]

The sides of the square is y₂ - y₁
*y₂ refers to the parabola
*y₁ refers to the line

ΔV = (y₁ - y₂)(y₁ - y₂)Δx
= (x - x²)(x - x²)dx

V = ₀∫¹ (x⁴ - 2x³ + x²)dx

etc..you will get 1/30

 

[.ben]

lol what does the diagram look like first?

 

[_ShiFTy_]

Just sketch the parabola and the line. Draw a strip perpendicular to the x axis (parallel to y axis). The length of this strip is one of the sides of the square. You can see that its the y value of the line minus the y value of the parabola. The area of the square will then be (y₁ - y₂)(y₁ - y₂)
The thickness of the strip is Δx...so the volume will be (y₁ - y₂)(y₁ - y₂)Δx

Just imagine thin rectangular prisms coming out of the page

 

[.ben]

oh ok but which way are the squares stacked?

 

[_ShiFTy_]

Rushed job...hope you get it now

 

[.ben]

oic, thanks shifty

 

[jarrypan]

oh.damn... the diagram is too complicated for me...I can think about it!