Wize University Calculus 2 Textbook > Bonus: Integral calculus in several variables (Videos Coming Soon)

Triple Integration (cylindrical)

Popular Courses

Cylindrical coordinates
Wize Concept
Cylindrical coordinates. A point P=(x,y,z)P = (x, y, z)P=(x,y,z) is represented by the ordered triple (r,θ,z)(r, \theta, z)(r,θ,z) where (r,θ)(r, \theta)(r,θ) are the polar coordinates in the xyxyxy-plane.
x=rcos⁡θr=x2+y2y=rsin⁡θtan⁡θ=yxz=zz=z\begin{array}{ll}
x=r\cos\theta & \hspace{3cm}r=\sqrt{x^2+y^2}\\[15pt]
y=r\sin\theta & \hspace{3cm}\tan\theta=\dfrac{y}{x}\\[15pt]
z=z & \hspace{3cm}z=z
\end{array}x=rcosθy=rsinθz=z​r=x2+y2​tanθ=xy​z=z​


→\to→ Notes:
Cylindrical coordinates are useful when there is symmetry about the zzz-axis (e.g. cylinders and cones).

Triple Integration (cylindrical)
Wize Concept
Triple Integration (cylindrical). Let D⊂R3D \subset \mathbb{R}^3D⊂R3 be defined by
g1(r,θ)≤z≤g2(r,θ),h1(θ)≤r≤h2(θ),a≤θ≤bg_1(r,\theta) \le z \le g_2(r, \theta),\quad h_1(\theta) \le r \le h_2(\theta),\quad a\le \theta \le bg1​(r,θ)≤z≤g2​(r,θ),h1​(θ)≤r≤h2​(θ),a≤θ≤b
where gig_igi​ and hih_ihi​ are continuous. If f(x,y,z)f(x, y, z)f(x,y,z) is continuous on DDD, then
∭Df(x,y,z)dV=∫ab∫h1(θ)h2(θ)∫g1(r)g2(r)f(rcos⁡θ,rsin⁡θ,z)rdzdrdθ.\iiint\limits_{D}f(x,y,z)dV=\int_a^b\int_{h_1(\theta)}^{h_2(\theta)}\int_{g_1(r)}^{g_2(r)}f(r\cos\theta,r\sin\theta,z)rdzdrd\theta.D∭​f(x,y,z)dV=∫ab​∫h1​(θ)h2​(θ)​∫g1​(r)g2​(r)​f(rcosθ,rsinθ,z)rdzdrdθ.

→\to→ Notes:
As with iterated integrals, the integral can be rearranged; take care to change the bounds of integration as well.
Don't forget the extra factor of rrr in the integrand.

Practice: Cylindrical coordinates
Represent the solid DDD bounded by z2=x2+y2z^2 = x^2 + y^2z2=x2+y2, x2+y2+z2=1x^2 + y^2 + z^2 = 1x2+y2+z2=1, and z≥0z\ge 0z≥0 using cylindrical coordinates.

∫02π∫012∫r1−r21 dz dr dθ\displaystyle \int_0^{2\pi}\int_0^{\frac{1}{\sqrt{2}}}\int_r^{\sqrt{1-r^2}}1\ dz\ dr\ d\theta∫02π​∫02​1​​∫r1−r2​​1 dz dr dθ

∫02π∫012∫r1−r2r dz dr dθ\displaystyle \int_0^{2\pi}\int_0^{\frac{1}{\sqrt{2}}}\int_r^{\sqrt{1-r^2}}r\ dz\ dr\ d\theta∫02π​∫02​1​​∫r1−r2​​r dz dr dθ

∫02π∫01∫r1−r21 dz dr dθ\displaystyle \int_0^{2\pi}\int_0^{1}\int_r^{\sqrt{1-r^2}}1\ dz\ dr\ d\theta∫02π​∫01​∫r1−r2​​1 dz dr dθ

∫02π∫01∫r1−r2r dz dr dθ\displaystyle \int_0^{2\pi}\int_0^{1}\int_r^{\sqrt{1-r^2}}r\ dz\ dr\ d\theta∫02π​∫01​∫r1−r2​​r dz dr dθ

None of the above

I don't know

Example: Triple integration (cylindrical)
Let DDD be the solid bounded by z2=x2+y2z^2 = x^2 + y^2z2=x2+y2, x2+y2+z2=1x^2 + y^2 + z^2 = 1x2+y2+z2=1, and z≥0z \ge 0z≥0. Evaluate 
∭DzdV\iiint\limits_{D}zdVD∭​zdV

Practice: Triple integration (cylindrical)
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

None of the above

I don't know

Extra Practice

Practice: Triple Integrals in Cylindrical Coordinates (1)

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)

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.