Evaluate the double integral by first identify it as the volume of a solid

Question

∫∫_R (4-2y)dA, R = [0, 1] x [0, 1]

Answer 1

To solve this problem one needs to know how the variable y is related to points in the domain R.

I will speculate that the domain R is in the x-y plane.

In that case, for each 0 ≤ x ≤ 1 and 0 ≤ y ≤ 1 the height of the solid is z=4-2y.

You can look at the solid from the point of view that its base is in the y-z-plane,

where it forms a parallelogram, and the solid's height in the x-direction is uniform 1.

The parallelogram has one side(along the z-axis) of length 4,

and the other parallel side of length 2.

The distance between the parallel sides is 1.

Therefore the area of the parallelogram is 1×(4+2)/2 = 3.

Multiplying the area of the parallelogram by the solid's height, which is 1,

we get a volume of 3.