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 $\sqrt{\cos x}$ and the x-axis, or simply $\sqrt{\cos x}$. 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

$A(x)=\dfrac12(\sqrt{\cos x})^2=\dfrac{\cos x}2$

Then the volume of this region would be

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