Ex..Findthevolumeofthetetrahedronboundedbythecoordinateplansasthe plan

1.	Ex..Findthevolumeofthetetrahedronboundedbythecoordinateplansasthe plane x+ y+ z1.
Answer» x+y+z=1 x+y=1 y=1-x (1-x-y)dy dx (Integral from 0 to 1 and Integral y= 0 to y=(1-x) V= (1-x-y) dy dx (y-xy-y^2/2)]0 to 1-x ((1-x)-x(1-x)-((1-x)^2)/2... dx (1-x)-(x+x^2)-((x^2-2x+1)... dx (1-x)-(x+x^2)-((x^2-2x+1)... dx (x^2-x+(1/2) dx (x^3/3) + (-x^2/2) + (x^3/3) ] 0 to 1 1/3 -1/2 +1/2 = 1/3 Answer: 1/3

Ex..Findthevolumeofthetetrahedronboundedbythecoordinateplansasthe plane x+ y+ z1.

Answer»

x+y+z=1

x+y=1

y=1-x

(1-x-y)dy dx

(Integral from 0 to 1 and Integral y= 0 to y=(1-x)

V= (1-x-y) dy dx

(y-xy-y^2/2)]0 to 1-x

((1-x)-x(1-x)-((1-x)^2)/2... dx

(1-x)-(x+x^2)-((x^2-2x+1)... dx

(x^2-x+(1/2) dx

(x^3/3) + (-x^2/2) + (x^3/3) ] 0 to 1

1/3 -1/2 +1/2 = 1/3

Answer: 1/3