Math Problem Statement

Hãy xác định cận cho các biến khi tính tích phân sau (không cần tính I): I = ∭ f(x, y, z) dx dy dz, với Ω là khối vật thể bị giới hạn bởi x^2 + y^2 + z^2 ≤ 4x x ≤ 2 − √(y^2 + z^2)

Solution

The problem involves determining the bounds for the triple integral over the volume Ω\Omega, which is bounded by the given constraints:

x2+y2+z24xx^2 + y^2 + z^2 \leq 4x x2y2+z2.x \leq 2 - \sqrt{y^2 + z^2}.

Step 1: Analyze the first constraint

The inequality x2+y2+z24xx^2 + y^2 + z^2 \leq 4x can be rewritten as: x24x+y2+z20.x^2 - 4x + y^2 + z^2 \leq 0. Complete the square for x24xx^2 - 4x: (x2)24+y2+z20,(x-2)^2 - 4 + y^2 + z^2 \leq 0, (x2)2+y2+z24.(x-2)^2 + y^2 + z^2 \leq 4. This represents a sphere centered at (2,0,0)(2, 0, 0) with radius 2.

Step 2: Analyze the second constraint

The inequality x2y2+z2x \leq 2 - \sqrt{y^2 + z^2} defines a region bounded by a vertical cylinder: x+y2+z22.x + \sqrt{y^2 + z^2} \leq 2. Square both sides to better understand the shape: x2+2xy2+z2+y2+z24.x^2 + 2x\sqrt{y^2 + z^2} + y^2 + z^2 \leq 4. This suggests a boundary involving a paraboloid shape (cylinder projection), but for bounds of integration, it can be interpreted in cylindrical or spherical coordinates as shown below.

Step 3: Use spherical or cylindrical coordinates for integration bounds

The bounds are easier to express in spherical coordinates: x=rcosϕsinθ, y=rsinϕsinθ, z=rcosθ.x = r \cos\phi \sin\theta,\ y = r \sin\phi \sin\theta,\ z = r \cos\theta.

Bounds in spherical coordinates:

  1. rr ranges from 00 to the sphere boundary (x2)2+y2+z2=4(x-2)^2 + y^2 + z^2 = 4.
  2. θ\theta (angle with the zz-axis) ranges from 00 to π\pi (full upper hemisphere).
  3. ϕ\phi (angle in the xyxy-plane) will depend on the intersection of the cylindrical constraint and the sphere.

Would you like me to explicitly compute and write all bounds in both spherical and cylindrical coordinates? Let me know!


Follow-up questions:

  1. How does the spherical constraint r2=4r^2 = 4 relate to the cylinder in Cartesian coordinates?
  2. Would the integration bounds differ if this were expressed only in cylindrical coordinates?
  3. How would the integral simplify if f(x,y,z)=1f(x, y, z) = 1?
  4. Can the intersection of the sphere and cylinder be visualized geometrically? How does it simplify integration?
  5. How do the transformations affect the limits of x,y,zx, y, z?

Tip: Converting complex 3D boundaries into spherical or cylindrical coordinates can significantly simplify triple integrals!

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Math Problem Analysis

Mathematical Concepts

Triple Integrals
Spherical Coordinates
Cylindrical Coordinates
Geometric Constraints

Formulas

x^2 + y^2 + z^2 ≤ 4x
(x-2)^2 + y^2 + z^2 ≤ 4

Theorems

Transformation to spherical or cylindrical coordinates

Suitable Grade Level

University level (Calculus III)