Question

Show that the area of the part of the plane z = ax + by + c that projects onto a region D in the xy-plane with area A(D) is squareroot a² + b² + 1 A(D).

Answer 1

The area of the part of the plane z = ax + by + c that projects onto a region D in the xy-plane is √(a² + b² + 1) A(D).

Step-by-step explanation:

To show that the area of the part of the plane z = ax + by + c that projects onto a region D in the xy-plane with area A(D) is √(a² + b² + 1) A(D), we can use the formula for the area of a parallelogram.

The area of a parallelogram with sides a and b is given by the magnitude of their cross product: |a × b|.

In this case, a = (1, 0, a) and b = (0, 1, b).

So, the magnitude of their cross product is |(1, 0, a) × (0, 1, b)| = √(a² + b² + 1).

Therefore, the area of the part of the plane z = ax + by + c that projects onto a region D in the xy-plane is √(a² + b² + 1) A(D).