Let V={(x,y,z):x^2+y^2 leq 4 and 0 leq z leq 4} be a cylinder and let P be the plane through (4, 0, 2),(0, 4, 2), and (-4, -4, 4). Compute the volume of C below the plane P.

Angel Kline 2022-10-12 Answered
Finding the volume in a cylinder that is intersected by a plane
I have a homework question that goes as follows.
Let V = { ( x , y , z ) : x 2 + y 2 4  and  0 z 4 } be a cylinder and let P be the plane through (4, 0, 2), (0, 4, 2), and (-4, -4, 4). Compute the volume of C below the plane P.
So I have these points and I set up my a ( x x 0 ) + b ( y y 0 ) + c ( z z 0 ) = 0 where ( x 0 , y 0 , z 0 ) is a point in the plane and ⟨a, b, c⟩ is perpendicular.
I then got two vectors that go between two points, namely:
P Q =< ( 0 4 , ( 4 0 ) , ( 2 , 2 ) >=< 4 , 4 , 0 > and P R =< ( 4 4 ) , ( 4 0 ) , ( 4 , 2 ) >=< 8 , 4 , 2 >
I then get the cross product of them to get < 8 , 8 , 48 > which is the coefficient in my equation of the plane:
8 ( x 4 ) + 8 ( y 0 ) + 48 ( z 2 ) = 0
I think that this is the equation of my plane. Is it?
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Answers (2)

wlanauee
Answered 2022-10-13 Author has 17 answers
Step 1
First of all, you made a mistake in your second vector when computing the direction of the plane:
P R = ( 8 , 4 , 2 )
and your cross product doesn't look correct even with your vector PR.
Now the volume can be set up like this:
S z d x d y
where S is the projection of the volume to the xy plane and z is the upper limit of z which is the plane represented by z.
Step 2
In fact, you will need to check whether the plane is below z = 4 in the confined region. After some trick you will see the plane is really below z = 4. That's why the z = 4 would not be in your integral any more.
The triple integral is 0 z d z d x d y
where z is exactly the plane. That is why I wrote the double integral.
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Ignacio Riggs
Answered 2022-10-14 Author has 4 answers
Step 1
If we let a = 4 , 4 , 0 and b = 8 , 4 , 2 , then a × b = 8 , 8 , 48 .
Then we can take n = 1 , 1 , 6 as a normal vector to the plane, so substituting in the coordinates of one of the points gives x + y + 6 z = 16.
Step 2
Now you can use polar coordinates or rectangular coordinates to find the volume,
since the base of the solid is the region in the xy-plane bounded by the circle x 2 + y 2 = 4,
and the height of the solid is given by z = 1 6 ( 16 x y )
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