+108 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)
Where it says cross-sections parallel to the y-axis, can you confirm that this really means cross-sections parallel to the yz plane ?
+108 For everyone who is helping me solve this problem, yes, the cross-sections are parallel to the yz-plane.
+128475 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
1
(sqrt (3) / 2 ) [ x^5 / 5 - (2/3)x^3 + x ] =
0
(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
0
