Practice: Triple integration (cylindrical)

Triple Integration (cylindrical)

5 Activities

Write down an integral using cylindrical coordinates representing ∭Eln⁡(x2+y2+z2)   ⁣dV\displaystyle\iiint_E\ln(x^2 + y^2 + z^2) \ \de{V}∭E​ln(x2+y2+z2)  dV  where EEE is the region inside the cylinder x2+y2=4x^2 + y^2 = 4x2+y2=4, outside the cylinder x2+y2=1x^2 + y^2 = 1x2+y2=1, and inside the sphere x2+y2+z2=9x^2 + y^2 + z^2 = 9x2+y2+z2=9.

∫04π∫12∫−9−r29−r2ln⁡(r2+z2)r2dzdrdθ\displaystyle\int^{4\pi}_0\int^2_1\int^{\sqrt{9−r^2}}_{−\sqrt{9−r^2}}\ln(r^2 + z^2)r^2dzdrd\theta∫04π​∫12​∫−9−r2​9−r2​​ln(r2+z2)r2dzdrdθ

∫02π∫12∫−9−r29−r2ln⁡(r2+z2)r2dzdrdθ\displaystyle\int^{2\pi}_0\int^2_1\int^{\sqrt{9−r^2}}_{−\sqrt{9−r^2}}\ln(r^2 + z^2)r^2dzdrd\theta∫02π​∫12​∫−9−r2​9−r2​​ln(r2+z2)r2dzdrdθ

∫04π∫12∫−9−r29−r2ln⁡(r2+z2)rdzdrdθ\displaystyle\int^{4\pi}_0\int^2_1\int^{\sqrt{9−r^2}}_{−\sqrt{9−r^2}}\ln(r^2 + z^2)rdzdrd\theta∫04π​∫12​∫−9−r2​9−r2​​ln(r2+z2)rdzdrdθ

∫02π∫12∫−9−r29−r2ln⁡(r2+z2)rdzdrdθ\displaystyle\int^{2\pi}_0\int^2_1\int^{\sqrt{9−r^2}}_{−\sqrt{9−r^2}}\ln(r^2 + z^2)rdzdrd\theta∫02π​∫12​∫−9−r2​9−r2​​ln(r2+z2)rdzdrdθ

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Evaluate ∫05∫−5−x20∫x2+y2−119−3x2−3y2(2x−3y)dzdydx\displaystyle\int_{0}^{\sqrt{5}}\int_{-\sqrt{5-x^2}}^{0}\int_{x^2+y^2-11}^{9-3x^2-3y^2}(2x-3y)dzdydx∫05​​∫−5−x2​0​∫x2+y2−119−3x2−3y2​(2x−3y)dzdydxusing cylindrical coordinates.

Practice: Triple integration (cylindrical)

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Triple Integration (cylindrical)

More Triple Integration (cylindrical) Questions:

Practice: Triple Integrals in Cylindrical Coordinates (1)