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Calculus IIChapter 6: 3-Dimensional SpaceSection 6-13

Spherical Coordinates — Question 6

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Question 6

Describe with spherical inequalities the solid inside the sphere ρ=4\rho=4 and above the cone ϕ=π/3\phi=\pi/3. Include the full azimuthal range and explain the meaning of “above.”

Original worksheet page 1: question and worked solution for 6-13-006
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Question 6 – Solution

Strategy A smaller polar angle lies closer to the positive zz-axis.

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Bounds Inside the sphere gives 0≤ρ≤40\le\rho\le 4. The full rotational sweep is 0≤θ<2π0\le\theta<2\pi. Above the cone means between the positive zz-axis and the conical surface, so 0≤ϕ≤π/30\le\phi\le\pi/3. Hence 0≤ρ≤4,0≤θ<2π,0≤ϕ≤π3.\boxed{0\le\rho\le 4,\qquad 0\le\theta<2\pi,\qquad 0\le\phi\le\frac{\pi}{3}}.

Boundary note At ρ=0\rho=0, angles are not unique, but including all angle values does not create additional geometric points.

Original worksheet page 2: question and worked solution for 6-13-006

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