Triple integrals in cylindrical coordinates practice problems
Triple integrals in cylindrical coordinates: dV = r dz dr dθ. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.
Use cylindrical coordinates to find \( \displaystyle \iiint_E z\, dV \) over the cylinder \( \displaystyle x^2 + y^2 \le 9 \), \( \displaystyle 0 \le z \le 2 \).
Use cylindrical coordinates to find the volume of the solid inside \( \displaystyle x^2 + y^2 = 9 \), above \( \displaystyle z = 0 \) and below the cone \( \displaystyle z = \sqrt{x^2 + y^2} \).
Use cylindrical coordinates to find the volume of the solid inside \( \displaystyle x^2 + y^2 = 4 \), above \( \displaystyle z = 0 \) and below \( \displaystyle z = 6 - x^2 - y^2 \).
Use cylindrical coordinates to find the volume of the solid inside \( \displaystyle x^2 + y^2 = 1 \), above \( \displaystyle z = 0 \) and below the cone \( \displaystyle z = \sqrt{x^2 + y^2} \).
Use cylindrical coordinates to find the volume of the solid inside \( \displaystyle x^2 + y^2 = 4 \), above \( \displaystyle z = 0 \) and below the cone \( \displaystyle z = \sqrt{x^2 + y^2} \).
Use cylindrical coordinates to find \( \displaystyle \iiint_E x^{2} + y^{2}\, dV \) over the cylinder \( \displaystyle x^2 + y^2 \le 1 \), \( \displaystyle 0 \le z \le 4 \).
Use cylindrical coordinates to find the volume of the solid inside \( \displaystyle x^2 + y^2 = 4 \), above \( \displaystyle z = 0 \) and below \( \displaystyle z = 4 - x^2 - y^2 \).
Use cylindrical coordinates to find \( \displaystyle \iiint_E z\, dV \) over the cylinder \( \displaystyle x^2 + y^2 \le 1 \), \( \displaystyle 0 \le z \le 4 \).
Use cylindrical coordinates to find \( \displaystyle \iiint_E z\, dV \) over the cylinder \( \displaystyle x^2 + y^2 \le 9 \), \( \displaystyle 0 \le z \le 3 \).
Use cylindrical coordinates to find \( \displaystyle \iiint_E z\, dV \) over the cylinder \( \displaystyle x^2 + y^2 \le 4 \), \( \displaystyle 0 \le z \le 3 \).