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.)

Answered: Volume Find the volume of the solid…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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 $z = 2 x^{2} + 2 y^{2}$ , Plane $z = 0$ , Cylinder $x^{2} + {(y - 1)}^{2} = 1$
To find: The volume of the solid bounded by given figures.

Step 2

We have,
Paraboloid $z = 2 x^{2} + 2 y^{2}$ , Plane $z = 0$
$\Rightarrow 0 \leq z \leq 2 x^{2} + 2 y^{2} \Rightarrow 0 \leq z \leq 2 (x^{2} + y^{2})$
Using the cylindrical coordinate system, we have-
$x^{2} + y^{2} = r^{2} \Rightarrow 0 \leq z \leq 2 r^{2}$
Now,
$x^{2} + {(y - 1)}^{2} = 1$
Take $x = r \cos θ, y = r \sin θ$ and substitute in the above equation, and we get-
$(r \cos θ) + {(r \sin θ - 1)}^{2} = 1 \Rightarrow r^{2} \cos^{2} θ + r^{2} \sin^{2} θ + 1 - 2 r \sin θ = 1 \Rightarrow r^{2} (\cos^{2} θ + \sin^{2} θ) + 1 - 2 r \sin θ = 1 \Rightarrow r^{2} (1) + 1 - 2 r \sin θ = 1 [∵ \cos^{2} θ + \sin^{2} θ = 1] \Rightarrow r^{2} = 2 r \sin θ \Rightarrow r = 2 \sin θ \Rightarrow 0 \leq r \leq 2 \sin θ$
Also, $0 \leq θ \leq 2 π$

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