What is the volume of the region bounded by y=sqrt(cosx) from [pi/2, pi/2] and whose cross sections are isosceles right triangles with horizontal leg in the xy
Mathematics
Santokat
Question
What is the volume of the region bounded by y=sqrt(cosx) from [pi/2, pi/2] and whose cross sections are isosceles right triangles with horizontal leg in the xy plane.
1 Answer

1. User Answers LammettHash
I assume the cross sections are taken perpendicular to the xaxis? This seems more likely than relative to the yaxis as far as easiness of calculation goes.
The base of each triangle is then determined by the distance between [tex]\sqrt{\cos x}[/tex] and the xaxis, or simply [tex]\sqrt{\cos x}[/tex]. Because it's a right triangle, you know the legs' lengths occur in a 1:1 ratio. Since each triangular cross section has one of its legs as its base, the heights must be the same as their bases.
So, the area of any one crosssection is
[tex]A(x)=\dfrac12(\sqrt{\cos x})^2=\dfrac{\cos x}2[/tex]
Then the volume of this region would be
[tex]\displaystyle\int_{\pi/2}^{\pi/2}\frac{\cos x}2\,\mathrm dx=1[/tex]