the base of a solid is a region in the first quadrant bounded by the x-axis, the y-axis, and the line y=1-x. if cross sections of the solid perpendicular to the x-axis are semicircles, what is the volume of the solid?

Answer 1

area of semicircle = (1/2) pi r^2 = (1/2) pi D^2/4 = (pi/8) D^2
I used D instead of r because the solid seems to be made up of circles sitting on this triangle with radius = y/2 and centered at y/2
So cross sectional area =(pi/8)y^2
so dV = (pi/8) dx [ (1-x)^2 ]
integrate from x = 0 to x = 1
(pi/8) [ 1 dx - 2 x dx + x^2 dx ]
=(pi/8) [ x -x^2 + (1/3)x^3 ] at x = 1 because all terms are 0 at x = 0
= (pi/8)(1/3)
= pi/24

Answer 2

Integrate pi(1-x)^2/2 dx from 0 to 1.
(let u = 1-x )
V = Integral pi u^2/2 du from 0 to 1
= pi/6

Each pi (1-x)^2 dx slab is a slab perpendicualr to the x axis

Answer 3

Damon is right. The individual slab area are pi (1-x)^2/8. I confused diameter with radius