bloodsky10
New member
- Joined
- Apr 2, 2017
- Messages
- 2
Hello everyone :-D
I'm studying Volume and multiple integrals theory. While doing some math, I get stuck with the shadow of intersection between plane and elliptic paraboloid (i guess) as follow:
A is the region described by { z ≥ 5x2 + 2y2 - 4xy , z ≤ x + 2y + 1 }. I need to calculate volume of this region |A|.
My first thought was to split volume triple integral like that :
∫∫∫dxdydz = ∫∫dxdy∫dz
Where the last integral in "dz" goes from 5x2 + 2y2 - 4xy to x + 2y + 1. Then switching to polar coordinates to integrate over the shadow region called D.
Placing 5x2 + 2y2 - 4xy = x + 2y + 1 gives me some problem. By using rapresentation tool, I ended up with and ellipse with center (1/2,1).
But how I can recognize it ? Tried some combination of u and v parameterization but can't figure it out.
How can I simplify this ?
Result from the book is : (27*pi*sqrt(6))/64
Thanks in advance for help.
I'm studying Volume and multiple integrals theory. While doing some math, I get stuck with the shadow of intersection between plane and elliptic paraboloid (i guess) as follow:
A is the region described by { z ≥ 5x2 + 2y2 - 4xy , z ≤ x + 2y + 1 }. I need to calculate volume of this region |A|.
My first thought was to split volume triple integral like that :
∫∫∫dxdydz = ∫∫dxdy∫dz
Where the last integral in "dz" goes from 5x2 + 2y2 - 4xy to x + 2y + 1. Then switching to polar coordinates to integrate over the shadow region called D.
Placing 5x2 + 2y2 - 4xy = x + 2y + 1 gives me some problem. By using rapresentation tool, I ended up with and ellipse with center (1/2,1).
But how I can recognize it ? Tried some combination of u and v parameterization but can't figure it out.
How can I simplify this ?
Result from the book is : (27*pi*sqrt(6))/64
Thanks in advance for help.
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