Use a triple integral to find the volume of the tetrahedron T bounded by the planes x+2y+z=2, x=2y, x=0 and z=0

Volume of the Tetrahedron T =

Step-by-step explanation:

As given, The tetrahedron T is bounded by the planes x + 2y + z = 2, x = 2y, x = 0, and z = 0

We have,

z = 0 and x + 2y + z = 2

⇒ z = 2 - x - 2y

∴ The limits of z are :

0 ≤ z ≤ 2 - x - 2y

Now, in the xy- plane , the equations becomes

x + 2y = 2 , x = 2y , x = 0 ( As in xy- plane , z = 0)

Firstly , we find the intersection between the lines x = 2y and x + 2y = 2

∴ we get

2y + 2y = 2

⇒4y = 2

⇒y = = 0.5

⇒x = 2() = 1

So, the intersection point is ( 1, 0.5)

As we have x = 0 and x = 1

∴ The limits of x are :

0 ≤ x ≤ 1

Also,

x = 2y

⇒y =

and x + 2y = 2

⇒2y = 2 - x

⇒y = 1 -

∴ The limits of y are :

≤ y ≤ 1 -

So, we get

Volume =

             =

             =

             =

             =

             =

             =

             = 1 - 1² + - 0 + 0 - 0

             = 1 - 1 + =  

So, we get

Volume =

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