Question

An object is in the first octant of a three-dimensional coordinate system and is bounded by the three axial planes and from above byz=1-x-y.

a) Find out wherez=1-x-y​​​​​​​ intersects the xy plane and outlines the base surface of the object in the xy plane.
b) Find the volume of the object by setting up and calculating a double integral.

Answer 1

To find where z=1-x-y intersects the xy plane, set z=0. Solve for x and y to get x+y=1, which represents a line in the xy plane. To find the base surface of the object in the xy plane, graph the line x+y=1. To find the volume of the object, set up a double integral to integrate over the region bounded by the three axial planes and z=1-x-y.

Step-by-step explanation:

To find where z=1-x-y intersects the xy plane, we need to set z equal to 0. So we have the equation 0=1-x-y.

Now, solving for x and y, we get x+y=1. This equation represents a line in the xy plane.

To find the base surface of the object in the xy plane, we can graph the line x+y=1.

To find the volume of the object, we can set up a double integral to integrate over the region bounded by the three axial planes and z=1-x-y. The integrand would be 1 (since each point in the region contributes a volume of 1), and the limits of integration would be determined by the equations of the axial planes and z=1-x-y.