Triple integrals in spherical coordinates practice problems
Triple integrals in spherical coordinates: dV = ρ² sin φ dρ dφ dθ. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Use spherical coordinates to find the volume of the upper half of the ball \( \displaystyle x^2 + y^2 + z^2 \le 4 \).
Use spherical coordinates to find the volume of the solid inside \( \displaystyle x^2 + y^2 + z^2 = 1 \) and above the cone \( \displaystyle z = \sqrt{x^2 + y^2} \).
Use spherical coordinates to find the volume of the solid inside \( \displaystyle x^2 + y^2 + z^2 = 9 \) and above the cone \( \displaystyle z = \sqrt{x^2 + y^2} \).
Use spherical coordinates to find \( \displaystyle \iiint_E \frac{dV}{\sqrt{x^2 + y^2 + z^2}} \) over the ball of radius 3.
Use spherical coordinates to find \( \displaystyle \iiint_E \frac{dV}{\sqrt{x^2 + y^2 + z^2}} \) over the ball of radius 2.
Use spherical coordinates to find \( \displaystyle \iiint_E z\, dV \), where \( \displaystyle E \) is the upper half of the ball \( \displaystyle x^2 + y^2 + z^2 \le 1 \).
Use spherical coordinates to find the volume of the upper half of the ball \( \displaystyle x^2 + y^2 + z^2 \le 1 \).
Use spherical coordinates to find \( \displaystyle \iiint_E (x^2 + y^2 + z^2)\, dV \) over the ball \( \displaystyle x^2 + y^2 + z^2 \le 1 \).
Use spherical coordinates to find \( \displaystyle \iiint_E (x^2 + y^2 + z^2)\, dV \) over the ball \( \displaystyle x^2 + y^2 + z^2 \le 4 \).
Use spherical coordinates to find the volume of the solid inside \( \displaystyle x^2 + y^2 + z^2 = 4 \) and above the cone \( \displaystyle z = \sqrt{x^2 + y^2} \).