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# Calc Help

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Hey guys! Bit stumped here. Hopefully, there is someone who can answer this on this website (ik its for less advanced math but I'm gonna ask anyways). Here it is:

Let S be the solid whose base is the region in the x, y-plane bounded by y=x^2 and y=1 whose cross-sections parallel to the y-axis are equilateral triangles. Find the volume of S. Thanks!

- BasicMaths (lol what a paradox)

Feb 11, 2021

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Where it says cross-sections parallel to the y-axis, can you confirm that this really means cross-sections parallel to the yz plane ?

Feb 12, 2021
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BasicMaths  Feb 12, 2021
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For everyone who is helping me solve this problem, yes, the cross-sections are parallel to the yz-plane.

Feb 15, 2021
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The  bounds on integration   are  from  x   -1  to  x =  1     (these are  the x intersections of the  graphs)

The base  of  each  triangle  =  ( 1 - x^2) =  the side of each triangle

The   height of  each  triangle   =  side *  sqrt (3) / 2

Thus.......the  area  for  each triangle   =  (1/2) base * ( height)   =

(1/2) ( 1  - x^2) (1  - x^2) *sqrt (3)  /2  =

(sqrt (3)  / 4)  ( 1  - x^2) ( 1  - x^2)  =   (sqrt (3)/ 4)   ( 1  - x^2)^2    =  (sqrt (3)  / 4) (x^4  - 2x^2  + 1)

And summing the  areas of  all the  triangles  from x  = -1   to  x = 1  will  give us the volume of S

Since  the  volume is symmetric   on each side of  the  y axis,  we can  actually  just  integrate  this

1

(2 * sqrt (3)  / 4)  ∫     x^4  -  2x^2   +  1    dx   =

0

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(sqrt (3) / 2 )  [ x^5 / 5   -  (2/3)x^3 +  x ]       =

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(sqrt (3)   / 2 )    [  1/5 - 2/3  +  1  ]  =

(sqrt (3) / 2)  [ 6/5 - 2/3 ] =

(sqrt (3)   / 2 )  [  18 - 10]   /15   =

(sqrt (3)   / 2  )  [ 8  / 15]  =

(4/15)sqrt (3)   ≈  .4619 units ^3   =  volume of  S

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Feb 17, 2021