Question

Find the volume of the tetrahedron in the first octant bounded by the coordinate planes and the plane passing through the point (4,0,0), (0,3,0) and (0,0,1).

Answer 1

Step1 Equation of plane in intercept form is (x/4)+(y/3)+(z/1)=1

Step2 let x/4=u; y/3 =v and z/1=w ; Jacobaian = 4x3x1=12

Step3 plane now becomes u+v+w=1

Step4 ; w varies from 0to 1-u-v ; v varies from0to 1-u and u varies from 0to 1

Step5 ; integrating with respect to v

Step6 ; put limits of v here from 0to 1-u

Step7 =

=- 1/6[1-u]^3

Step8 Value of integral at upper limit =0 ; At lower limit = -1/6

Step9 Value of integral= 0-(-1/6)= 1/6

Step10 Volume = Jacobian x1/6 =12x1/6=2 units