FInd the vol. of a general prismatoid whose height is 2, and for which A sub.y=15+5y-3y^2, where y is the distnce from the base of the prismatoid.
This is from a book of Kern and Bland Solid Mensuration Chapter 8, page 125 no.1 Can ou aslo help me with the other nubers, if you can.
The two parallel planes of this prismatoid is lieing on y=0 plane and on y=2 plane.
The volume of this can be found by using simple integration,
consider a plane at y with dy width. The area of that palne is,
A = 15+5y-3y^2
The volume of the plane can be approximated to,
dV = A*dy
dV = (15+5y-3y^2)dy
Now if we integrate this from y=0 to y=2, we can find the volume of the prismatoid.
`V = intdV = intAdy = int_0^2(15+5y-3y^2)dy`
Evaluating the integral,
int(15+5y-3y^2)dy = (15y+(5y^2)/2-y^3
`int_0^2(15+5y-3y^2)dy = (15*2+(5*2^2)/2-2^3)-(15*0+(5*0^2)/2-0^3)`
`int_0^2(15+5y-3y^2)dy = 30+20 -8 = 42`
Therefore the volume of the prismatoid is 42.