Question

CALCULUS 3.

Having Difficulty setting up.
Use a double integral to find the exact volume of the wedge cut from the
cylinder x^2+y^2=4 by the planes z=0 and z=2-y

Thank you.

Answer 1

draw the region first. ALWAYS draw the region if you can
You can either choose to draw the domain of integration, or draw the entire volume. Up to you.

Answer 2

That where the problme is.

Answer 3

What's the issue?

Answer 4

I know Z=0 is the XY plane

Answer 5

and?

Answer 6

Z=2-y is a straight line.

Answer 7

is it a straight line?

Answer 8

what values can x take?

Answer 9

you mean for the cylinder.

Answer 10

No. For z=2-y

Answer 11

but there is no x in the equation, x=0

Answer 12

oh? why can't x be equal to 1? Is the equation true if x=0? if x=1? how about x=2?

Answer 13

The equation is true for x=0, because the line is on the yz plane. x cannot take any value.

Answer 14

??

Answer 15

Is there anything in the equation saying that x has to be equal to 0?

Answer 16

Nope.

Answer 17

Then how do you know?

Answer 18

Why are we talking about x here.

Answer 19

we don't have x.

Answer 20

Because, you cannot do the problem without your domain of integration. you assume that z=2-y graphs a straight line, but how are you supposed to do the problem in R^3 when you have a line, a plane, and a cylinder? surely one of the assumptions you have made is wrong.

Answer 21

let's look at z=0
what does that graph?
what values can y, and x take on?

Answer 22

Z=0 y=2

Answer 23

no, I'm talking about what you said was the xy plane. z=0

Answer 24

its a plane xy-plane

Answer 25

yes. What values can x take? what values can y take?

Answer 26

According to your logic for z=2-y, x=0 and y=0 too.

Answer 27

z=0 is the whole xy plane both x and y can have any value

Answer 28

well at least positive

Answer 29

then what is z=2-y?

Answer 30

on 2D or 3D

Answer 31

and are you sure x and y have to be positive? we are talking about a plane, not a quarter plane

Answer 32

What do you think the problem is referring to? 2D or 3D, you judge

Answer 33

I think everything here is in 3D?

Answer 34

but not for the domain

Answer 35

so what is z=2-y?

Answer 36

A Plane

Answer 37

so, what does the domain look like?

Answer 38

from 0 to the plane z=y-2

Answer 39

that is for z yes. what about x and y?

Answer 40

I am totally
lost

Answer 41

Okay. So the first thing we always do is sketch the region, or the domain we integrate in, no? can you do this?

Answer 42

I cannot sketch the region, and cant figure the domain either.

Answer 43

when you say region you mean the cylinder?

Answer 44

No. We are doing a double integral here, so we don't actually integrate the cylinder with a density function of one. We integrate a CUT of the cylinder that is more or less of constant cross sectional area... take your guess which plane we cut the cylinder in

Answer 45

the cylinder is on the z axis and the plane z=2-y cut the cylinder ?

Answer 46

No. we actually integrate over the region x^2+y^2=4

Answer 47

Oh yeah I know that.

Answer 48

The headache is finding the domain?

Answer 49

which i cannot still figure out.

Answer 50

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fill in what you know