PLS HELP I NEED THIS DONE ASAP AND IT IWLL MEAN ALOT AND ILL MAKRK U AS BRAINLIST ITS 6TH GRADE GEOGRAPHY

The area of the shaded polygon is 31 square units.

Step-by-step explanation:

The procedure to calculate the area of the shaded irregular polygon is described below:

1) Calculate the area of the initial rectangle.

2) Calculate the areas of the four triangles.

3) Subtract the areas of the four triangles from the initial triangle.

The area of the shaded polygon is 31 square units.

Parameterize the surface (I'll call it S) by

r(u, v) = (1 - u) (1 - v) i + u (1 - v) j + v k

with 0 ≤ u ≤ 1 and 0 ≤ v ≤ 1.

Take the normal vector to this surface to be

n = ∂r/∂u × ∂r/∂v = ((v - 1) i + (1 - v) j) × ((u - 1) i - u j + k) = (1 - v) (i + j + k)

with magnitude

||n|| = √3 (1 - v)

Then in the integral, we have

Alternatively, if you're not familiar with parameterizing surfaces, you can use the "projection" formula:

where I write to mean the projection of the surface onto the (x, y)-plane, and z = f(x, y). We would then use

x + y + z = 1   ==>   z = f(x, y) = 1 - x - y

and is the triangle,

{(x, y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 - x}

Then the integral becomes