1) Find the area of the part of the plane4x + 3y + z = 12that lies in the first octant.2) Use polar coordinates to find the volume of the given solid.Bounded by the paraboloid z = 5 + 2x^2 + 2y^2 and the plane z = 11 in the first octant

Emeli Hagan

Emeli Hagan

Answered question

2021-06-08

1) Find the area of the part of the plane
4x+3y+z=12
that lies in the first octant.
2) Use polar coordinates to find the volume of the given solid.
Bounded by the paraboloid z=5+2x2+2y2 and the plane z = 11 in the first octant

Answer & Explanation

aprovard

aprovard

Skilled2021-06-09Added 94 answers

I have resolved your question. The answer is below:

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Jeffrey Jordon

Jeffrey Jordon

Expert2021-09-30Added 2605 answers

1) 4x+3y+3=12

R(x,y)=<x,y,124x3y>

|Rx×Ry|=(fx)2+(fy)2+1=(4)2+(3)2+1

=16+9+1=26

Sds=D26dA

(x,y,3) axis =(3,0,0)(0,4,0)(0,0,12)

So, the domain D in triangle is bounded by (0,0)(3,0)(0,4)

D26dA=26×12×3×42=626

z=5+2x2+2y2

11=5+2x2+2y2

6=2x2+2y2x2+y2=3 |x2+y2=R2

Volume =0π203(5+2x2+2y2)dA

=0π203(5+2R)RdRdQ=0π2033R+2R2drdQ

=0π2[5R28+2R33]03dQ=0π2152+63dQ

15π4+3π=π[154+3] Answer

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