Volume of solid by using projection to different planes

[chetzread]

Homework Statement

Find the volume of solid which is bounded by z = 4-x-y and below by region in the plane of 0<x<2 , 0<y<1
When i use zx -plane projection , i found that my ans is different with the ans of using xy projection ...Which part i did wrongly ?

From the ans given , volume = 5 unit , when i use xy plane projection , i gt this ans , but when I use zx -plane projection , i got V = 6

For zx -plane projection , i got V = ∫∫∫ dydxdz , 0<x <2 , 0 <z< 4-x , 0<y<1

Homework Equations

The Attempt at a Solution

 

[LCKurtz]

Homework Statement

Find the volume of solid which is bounded by z = 4-x-y and below by region in the plane of 0<x<2 , 0<y<1
When i use zx -plane projection , i found that my ans is different with the ans of using xy projection ...Which part i did wrongly ?

From the ans given , volume = 5 unit , when i use xy plane projection , i gt this ans , but when I use zx -plane projection , i got V = 6

For zx -plane projection , i got V = ∫∫∫ dydxdz , 0<x <2 , 0 <z< 4-x , 0<y<1

Homework Equations

The Attempt at a Solution

Your zx plane projection is incorrect. You need two integrals because when you integrate in the y direction first, sometimes you hit the plane $y=1$ and sometimes you hit the plane $z = 4-x-y$. In fact if you do the order dydxdz instead of dydzdx you may need 3 integrals.

 

[chetzread]

Your zx plane projection is incorrect. You need two integrals because when you integrate in the y direction first, sometimes you hit the plane $y=1$ and sometimes you hit the plane $z = 4-x-y$. In fact if you do the order dydxdz instead of dydzdx you may need 3 integrals.

sorry , i still didnt get you , how should the whole integral look like ? What is the limit fro x , y and z ?

 

[LCKurtz]

sorry , i still didnt get you , how should the whole integral look like ? What is the limit fro x , y and z ?

I'm not going to work it for you. What you need to do is imagine you are looking at the zx plane from straight down the y axis. Take your picture and project every line onto the zx plane. That will show you the shape of the zx region, which is what you need to find the limits.

 

[chetzread]

I'm not going to work it for you. What you need to do is imagine you are looking at the zx plane from straight down the y axis. Take your picture and project every line onto the zx plane. That will show you the shape of the zx region, which is what you need to find the limits.

is my diagram wrong ?

 

[chetzread]

here's my zx plane . with 0<x <2 , 0 <z< 4-x , is it correct ?

 

[LCKurtz]

Here's a correct picture:

You need to draw the projection of $a$ and $b$ and the line between them on the zx plane. The point $a$ is incorrectly drawn in your original sketch