Volume Find the volume of the solid bounded by the paraboloidz = 2x2 + 2y2, the plane z = 0, and the cylinderx2 + (y - 1)2 = 1. (Hint: Use symmetry.)

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
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Volume Find the volume of the solid bounded by the paraboloid
z = 2x2 + 2y2, the plane z = 0, and the cylinder
x2 + (y - 1)2 = 1. (Hint: Use symmetry.)

Expert Solution
Step 1

Given: Paraboloid z=2x2+2y2, Plane z=0, Cylinder x2+y-12=1
To find: The volume of the solid bounded by given figures.

Step 2

We have,
Paraboloid z=2x2+2y2, Plane z=0
0z2x2+2y20z2x2+y2
Using the cylindrical coordinate system, we have-
x2+y2=r20z2r2
Now,
x2+y-12=1
Take x=r cosθ, y=r sinθ and substitute in the above equation, and we get-
    r cosθ+r sinθ-12=1r2 cos2θ+r2 sin2θ+1-2r sinθ=1r2 cos2θ+sin2θ+1-2r sinθ=1r21+1-2r sinθ=1     cos2θ+sin2θ=1r2= 2r sinθr= 2 sinθ0r2 sinθ
Also, 0θ2π

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