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.
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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.
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