Question

Let f(x,y) = x² + y2 +3 and let R be the region in the xy-plane bounded above by
y=-x?+ 4 and below by y=-5. Calculateſſ f(x,y) dA:
f(x,y) da
R
X х
90 - 10x² + 3x
is a function that sweeps​

Answer 1

Answer:

Let f(x,y) be a differentiable function. As we have seen, z=f(x,y) defines a surface in xyz-space. In some applications, it necessary to know the surface area of the surface above some region R in the xy-plane. See the figure.

The formula for the surface area is

displaymath58

This is a double integral.

Example

What is the surface area of the plane z=2x+3y above the rectangle with -1<=x<=2 and 0<=y<=2? In this case f_x=2 and f_y=3. Applying the above formula, the surface area S is given by

displaymath60

Since, the region of integration R is a rectangle and the integrand is continuous, the value of the integral is independent of the order of integration. It can be shown that S=6*sqrt(14).

Example

Find the surface area of the part of the paraboloid z=16-x^2-y^2 that lies above the xy plane (see the figure below).

The region R in the xy-plane is the disk 0<=x^2+y^2<=16 (disk or radius 4 centered at the origin).

For this problem, f_x=-2x and f_y=-2y.