if four points are collinear, that are also coplanar: This is true
If points lie in the same line. they must lie in the same plane
Note the coordinates of each point: R(-4, 5), S(5, 1), T(2, -3).
The centroid is the point whose coordinates are the average of the coordinates of R, S, and T.
<em>x</em>-coordinate: (-4 + 5 + 2)/3 = 3/3 = 1
<em>y</em>-coordinate: (5 + 1 - 3)/3 = 3/3 = 1
So the centroid is (1, 1).
It is hard to comprehend your question. As far as I understand:
f(x,y) = e^(-x)
Find the volume over region R = {(x,y): 0<=x<=ln(6), -6<=y <= 6}.
That is all I understood. It would be easier to understand with a picture or some kind of visual aid.
Anyways, to find the volume between the surface and your rectangular region R, we must evaluate a double integral of f on the region R.

Now evaluate,

which evaluates to, 5/6 if I did the math correct. Correct me if I am wrong.
Now integrate this w.r.t. y:

So,

Your answer would be 10.6
Make the circle closed and the > needs to be underlined