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.

Answer 1

I assume the cross sections are taken perpendicular to the x-axis? This seems more likely than relative to the y-axis as far as easiness of calculation goes.

The base of each triangle is then determined by the distance between
`√(\cos x)` and the x-axis, or simply
`√(\cos x)`. 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 cross-section is


`A(x)=\frac12(√(\cos x))^2=\frac{\cos x}2`

Then the volume of this region would be


`\displaystyle\int_(-\pi/2)^(\pi/2)\frac{\cos x}2\,\mathrm dx=1`