Integrate over the region enclosed by z = 1 + x + y and the xy plane

[Cloudless]

1. ∫∫∫_E6xy dV, where E lies under the plane z = 1 + x + y

Apparently according to a classmate, the limits are:

-1≤x≤0, -1-x≤y≤0, and 0≤z≤ 1+x+y

I know how to get z. But I am confused for y and x.

To solve the limits of y, you would plug in 0 for z, getting y = -1 - x. But why is the upper bound 0?

Same for x. I assume you plug in 0 for both y and z when solving for x, resulting in x = -1. I am not sure why the upper bound is 0.

 

[LCKurtz]

1. ∫∫∫_E6xy dV, where E lies under the plane z = 1 + x + y

Apparently according to a classmate, the limits are:

-1≤x≤0, -1-x≤y≤0, and 0≤z≤ 1+x+y

I know how to get z. But I am confused for y and x.

To solve the limits of y, you would plug in 0 for z, getting y = -1 - x. But why is the upper bound 0?

Same for x. I assume you plug in 0 for both y and z when solving for x, resulting in x = -1. I am not sure why the upper bound is 0.

As stated, the problem has no finite answer because as x and y increase, so does z and there is an infinite region "under" the plane. So the problem probably mentions that not just under the plane but probably limited by the coordinate planes. Assuming that, my advice to you is to draw a picture. Plot the 3 intercepts of the plane giving a triangular portion and look at the xy domain for that part of the plane. That's where you will find the limits.