Use a double integral to find the volume of the solid in the first octant bounded by the plane determined by the three non-colinear points P(3, 0, 0), Q(0, 1, 0),R(0, 0, 2). Draw a picture of the volume you are finding.

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:

- 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:

step-by-step explanation:

p = 6s

step-by-step explanation: