Volume help?????????????

Question

http://i38.tinypic.com/2j2w32d.jpg

Determine the way slice the water into parallel rectangular cross-sections and then set up a definite integral for the volume of the water in the glass.

Answer 1

if they want the cross sections to be rectangular, the slices are actually perpendicular to the base (meaning if the 'blue' solid is resting on the table, the slices are vertical) the slices are also perpendicular to that specific diameter whose one end is the intersection of two surfaces - the bottom and the top surfaces.

the width of the rectangle simply follows the circle of the base

x^2 + y^2 = r^2

y = √(r^2 - x^2) ... x from -r to r.

the height follows the line such that if x = -r, y = 0, if x = r, y = L

y = mx + b

0 = -rm + b

L = rm + b

b = L/2

m = L/2r

thus

y = (L/2r)x + (L/2)

thus

V = ∫(-r , r) width*height dx