Finding Volume by Cross Section Area

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Discussion Overview

The discussion revolves around finding the volume of a solid formed by the curves y=sinx and y=sinx + 2, bounded by x=4 and x=-4, using triangular cross sections. The focus is on determining the appropriate formula for the area of the cross sections and setting up the integral for volume calculation.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant expresses uncertainty about how to start finding the volume and the formula for the area of the cross section.
  • Another participant questions what kind of triangular cross sections are being used and suggests that a sketch could help clarify the situation.
  • It is established that the cross sections are equilateral triangles, but there is confusion about the formula for the base of the triangle and how to apply it in the integral.
  • One participant proposes that the base of the triangle is the positive difference between the values of the two curves at position x, noting that care is needed outside the interval (0, π) due to the curves crossing each other.
  • A later reply indicates agreement with the previous point but expresses uncertainty about the specifics of the calculations.

Areas of Agreement / Disagreement

Participants generally agree on the shape of the cross sections being equilateral triangles and the need to find the base as the difference between the curves. However, there remains uncertainty regarding the specific calculations and the setup of the integral.

Contextual Notes

There are unresolved questions about the exact values to plug into the formulas and how to handle the intervals where the curves intersect.

heather7113
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I need to find the volume of this figure y=sinx, y=sinx + 2, x=4, x=-4 using the area of triangular cross sections. i don't really know where to start...how to find the formula for area of cross section etc. HELP!
 
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heather7113 said:
I need to find the volume of this figure y=sinx, y=sinx + 2, x=4, x=-4 using the area of triangular cross sections. i don't really know where to start...how to find the formula for area of cross section etc. HELP!

What kind of triangle cross sections? Right? Equilateral? Have you drawn a sketch of this? Do you know the formula for the area of a triangle? A sketch will help.
 
The cross sections are equilateral triangles. And I have drawn a sketch - I'm just not sure what the integral should be - like what is the formula for the base of the triangle. I know the formula is sgrt 3 x b / 4 I just don't really know what to plug in and what the numbers should be...?
 
heather7113 said:
The cross sections are equilateral triangles. And I have drawn a sketch - I'm just not sure what the integral should be - like what is the formula for the base of the triangle. I know the formula is sgrt 3 x b / 4 I just don't really know what to plug in and what the numbers should be...?

Isn't the base of the triangle just the positive difference between the values of the two curves at position x? And the slice has thickness dx. At least that is so on the interval (0,\pi). Outside of that interval you have to be careful because the curves cross each other at 0 and \pi.
 
Thanks. Thats what I was thinking, I just wasn't sure...
 

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