Answer:
2.5 units^3
Step-by-step explanation:
Given:-
- The solid is bounded by a plane defined by the following points:
                        P(3, 0, 0), Q(0, 1, 0),R(0, 0, 2)
Find:-
Use a double integral to find the volume of the solid in the first octant bounded by the plane
Solution:-
- Determine the equation of the plane. Compute two direction vectors d1 and d2 that lie on the plane:
                               d1 = P - Q
                               d1 = (3, 0, 0) - (0, 1, 0) = (3,-1,0)
                               d2 = P - R
                               d2 = (3, 0, 0) - (0, 0, 2) = (3,0,-2) 
- Find the a vector "normal" - n to the plane by cross product formulation of direction vectors (d1 and d2) that lie on the plane:
                ![n = d1xd2 = \left[\begin{array}{c}3&-1&0\end{array}\right] x \left[\begin{array}{c}3&0&-2\end{array}\right] = \left[\begin{array}{ccc}3&-1&0\\3&0&2\end{array}\right] = \left[\begin{array}{c}-2&-6&3\end{array}\right]](https://tex.z-dn.net/?f=n%20%3D%20d1xd2%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D3%26-1%260%5Cend%7Barray%7D%5Cright%5D%20x%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D3%260%26-2%5Cend%7Barray%7D%5Cright%5D%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bccc%7D3%26-1%260%5C%5C3%260%262%5Cend%7Barray%7D%5Cright%5D%20%3D%20%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D-2%26-6%263%5Cend%7Barray%7D%5Cright%5D) 
 
- The equation of plane is:
                                 n.(x,y,z) = n.P
                                -2x -6y + 3z = -6
- The function of one variable would be:
                                 z = (2/3)x + 2y - 2 
- The double integration formulation would be:
                                 
- Where the limits (c and d) are defined by planar (x-y) projection of plane (n) :
                               y = d = -(1/3)x + 1 
                               c = 0
- Evaluate the limits:
                               