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 cylindrical coordinates partial integrals double integrals triple integrals (rectangular)

### Trig Identities and Formulas

basic trig identities

$$\sin^2\theta+\cos^2\theta=1$$   |   $$1+\tan^2\theta=\sec^2\theta$$

$$\displaystyle{\tan\theta=\frac{\sin\theta}{\cos\theta}}$$   |   $$\displaystyle{\cot\theta=\frac{\cos\theta}{\sin\theta}}$$

$$\displaystyle{\sec\theta=\frac{1}{\cos\theta}}$$   |   $$\displaystyle{\csc\theta=\frac{1}{\sin\theta}}$$

power reduction (half-angle) formulae

$$\displaystyle{\sin^2\theta=\frac{1-\cos(2\theta)}{2}}$$   |   $$\displaystyle{\cos^2\theta=\frac{1+\cos(2\theta)}{2}}$$

double angle formulae

$$\sin(2\theta)=2\sin\theta\cos\theta$$   |   $$\cos(2\theta)=\cos^2\theta-\sin^2\theta$$

list of trigonometric identities - wikipedia

trig sheets - pauls online notes

17calculus trig formulas - full list

### Calculus Topics Listed Alphabetically

Single Variable Calculus

 Absolute Convergence Alternating Series Arc Length Area Under Curves Chain Rule Concavity Conics Conics in Polar Form Conditional Convergence Continuity & Discontinuities Convolution, Laplace Transforms Cosine/Sine Integration Critical Points Cylinder-Shell Method - Volume Integrals Definite Integrals Derivatives Differentials Direct Comparison Test Divergence (nth-Term) Test
 Ellipses (Rectangular Conics) Epsilon-Delta Limit Definition Exponential Derivatives Exponential Growth/Decay Finite Limits First Derivative First Derivative Test Formal Limit Definition Fourier Series Geometric Series Graphing Higher Order Derivatives Hyperbolas (Rectangular Conics) Hyperbolic Derivatives
 Implicit Differentiation Improper Integrals Indeterminate Forms Infinite Limits Infinite Series Infinite Series Table Infinite Series Study Techniques Infinite Series, Choosing a Test Infinite Series Exam Preparation Infinite Series Exam A Inflection Points Initial Value Problems, Laplace Transforms Integral Test Integrals Integration by Partial Fractions Integration By Parts Integration By Substitution Intermediate Value Theorem Interval of Convergence Inverse Function Derivatives Inverse Hyperbolic Derivatives Inverse Trig Derivatives
 Laplace Transforms L'Hôpital's Rule Limit Comparison Test Limits Linear Motion Logarithm Derivatives Logarithmic Differentiation Moments, Center of Mass Mean Value Theorem Normal Lines One-Sided Limits Optimization
 p-Series Parabolas (Rectangular Conics) Parabolas (Polar Conics) Parametric Equations Parametric Curves Parametric Surfaces Pinching Theorem Polar Coordinates Plane Regions, Describing Power Rule Power Series Product Rule
 Quotient Rule Radius of Convergence Ratio Test Related Rates Related Rates Areas Related Rates Distances Related Rates Volumes Remainder & Error Bounds Root Test Secant/Tangent Integration Second Derivative Second Derivative Test Shifting Theorems Sine/Cosine Integration Slope and Tangent Lines Square Wave Surface Area
 Tangent/Secant Integration Taylor/Maclaurin Series Telescoping Series Trig Derivatives Trig Integration Trig Limits Trig Substitution Unit Step Function Unit Impulse Function Volume Integrals Washer-Disc Method - Volume Integrals Work

Multi-Variable Calculus

 Acceleration Vector Arc Length (Vector Functions) Arc Length Function Arc Length Parameter Conservative Vector Fields Cross Product Curl Curvature Cylindrical Coordinates
 Directional Derivatives Divergence (Vector Fields) Divergence Theorem Dot Product Double Integrals - Area & Volume Double Integrals - Polar Coordinates Double Integrals - Rectangular Gradients Green's Theorem
 Lagrange Multipliers Line Integrals Partial Derivatives Partial Integrals Path Integrals Potential Functions Principal Unit Normal Vector
 Spherical Coordinates Stokes' Theorem Surface Integrals Tangent Planes Triple Integrals - Cylindrical Triple Integrals - Rectangular Triple Integrals - Spherical
 Unit Tangent Vector Unit Vectors Vector Fields Vectors Vector Functions Vector Functions Equations

Differential Equations

 Boundary Value Problems Bernoulli Equation Cauchy-Euler Equation Chebyshev's Equation Chemical Concentration Classify Differential Equations Differential Equations Euler's Method Exact Equations Existence and Uniqueness Exponential Growth/Decay
 First Order, Linear Fluids, Mixing Fourier Series Inhomogeneous ODE's Integrating Factors, Exact Integrating Factors, Linear Laplace Transforms, Solve Initial Value Problems Linear, First Order Linear, Second Order Linear Systems
 Partial Differential Equations Polynomial Coefficients Population Dynamics Projectile Motion Reduction of Order Resonance
 Second Order, Linear Separation of Variables Slope Fields Stability Substitution Undetermined Coefficients Variation of Parameters Vibration Wronskian

### Search Practice Problems

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17calculus > partial integrals > triple integrals (cylindrical)

 Getting Started Describing Surfaces and Volumes Setting Up and Evaluating Practice

On this page we cover triple integrals in cylindrical coordinates and several applications. Triple integrals in rectangular coordinates are covered on a separate page, as well as triple integrals in spherical coordinates.
Understanding of cylindrical coordinates is necessary for understanding the material on this page.

Describing Surfaces and Volumes in Cylindrical Coordinates

So, you may ask, why do we need the cylindrical coordinate system when we describe surfaces and volumes in the rectangular coordinate system? Well, there are two reasons.
1. Some surfaces and volumes are more easily (simply) described in cylindrical coordinates. An example is given below.
2. When we get to triple integrals, some integrals are more easily evaluated in cylindrical coordinates and you will even have some integrals that can't be evaluated in rectangular coordinates but can be in cylindrical.

Setting Up and Evaluating Triple Integrals in Cylindrical Coordinates

Let's start by watching this short video clip, explaining cylindrical coordinates again and showing how to set up triple integrals in cylindrical coordinates.

### MIP4U - Triple Integrals Using Cylindrical Coordinates [2mins-7secs]

video by MIP4U

A couple of clarifications are in order that he mentions in this video.
First, he mentions that there is an extra factor of r in the term $$dV$$ in cylindrical coordinates but he doesn't explain where it comes from. Remember from your study of polar coordinates that $$dA = dx~dy$$ in rectangular coordinates becomes $$dA = r~dr~d\theta$$. This next video clip explains where $$dV$$ comes from in more detail.

### Larson Calculus - Triple Integrals in Cylindrical Coordinates [5mins-26secs]

This video will not stop automatically at the 5min-26sec mark. For the purposes of the current discussion, you can stop it there. However, there is some good content after that point that is okay to watch if you have time.

video by Larson Calculus

The second clarification involves the order of components of $$dV$$. In the first video, he says that $$dV = r~dz~dr~d\theta$$. This is only one of six possible representations for $$dV$$. The order is determined by how the volume is described by the equations. To determine what order to integrate when setting up the integral, we use the same idea as we did when setting up double integrals.

In integral form, a triple integral in cylindrical coordinates looks like this.

$$\displaystyle{ \iiint\limits_V {f(x,y,z) ~ dV} = }$$ $$\displaystyle{ \iiint\limits_V{ f(r\cos\theta, r\sin\theta,z)~r~dr~d\theta~dz } }$$

VERY IMPORTANT NOTE - - - Do not forget the $$r$$ in $$\color{red}{r}~dr~d\theta~dz$$ in the above equation. This is the most common mistake made by students learning this technique. Why do we have this extra $$r$$? Here is a video explaining this.

### Michel vanBiezen - Cylindrical Coordinates: Small Volume Element dV [4mins-0secs]

video by Michel vanBiezen

Before going on to spherical coordinates, work some practice problems involving cylindrical coordinates.

### Practice

Conversion Between A-B-C Level (or 1-2-3) and New Numbered Practice Problems

Please note that with this new version of 17calculus, the practice problems have been relabeled but they are MOSTLY in the same order. So, Practice A01 (1) is probably the first basic practice problem, A02 (2) is probably the second basic practice problem, etc. Practice B01 is probably the first intermediate practice problem and so on.

GOT IT. THANKS!

Instructions - Unless otherwise instructed, evaluate these integrals in cylindrical coordinates.

Basic Problems

Integrate $$f(x,y,z) = x^2 + y^2$$ over the solid region bounded by $$z = 0$$ and $$z = 4 - \sqrt{x^2+y^2}$$ in cylindrical coordinates.

Problem Statement

Integrate $$f(x,y,z) = x^2 + y^2$$ over the solid region bounded by $$z = 0$$ and $$z = 4 - \sqrt{x^2+y^2}$$ in cylindrical coordinates.

$$512 \pi/5$$

Problem Statement

Integrate $$f(x,y,z) = x^2 + y^2$$ over the solid region bounded by $$z = 0$$ and $$z = 4 - \sqrt{x^2+y^2}$$ in cylindrical coordinates.

Solution

### 1929 video

video by MIP4U

$$512 \pi/5$$

$$\displaystyle{ \int_{-2}^{2}{ \int_{-\sqrt{4-y^2}}^{ \sqrt{4-y^2}}{\int_{\sqrt{x^2+y^2}}^{2}{ xz } } } }$$ $$dz ~dx ~dy$$

Problem Statement

$$\displaystyle{ \int_{-2}^{2}{ \int_{-\sqrt{4-y^2}}^{ \sqrt{4-y^2}}{\int_{\sqrt{x^2+y^2}}^{2}{ xz } } } }$$ $$dz ~dx ~dy$$

0

Problem Statement

$$\displaystyle{ \int_{-2}^{2}{ \int_{-\sqrt{4-y^2}}^{ \sqrt{4-y^2}}{\int_{\sqrt{x^2+y^2}}^{2}{ xz } } } }$$ $$dz ~dx ~dy$$

Solution

### 1968 video

video by Krista King Math

0

Calculate the volume of the solid bounded by $$z = 0$$ and $$z = 9 - x^2 - y^2$$.

Problem Statement

Calculate the volume of the solid bounded by $$z = 0$$ and $$z = 9 - x^2 - y^2$$.

$$81 \pi/2$$

Problem Statement

Calculate the volume of the solid bounded by $$z = 0$$ and $$z = 9 - x^2 - y^2$$.

Solution

### 1970 video

video by MIP4U

$$81 \pi/2$$

Calculate the volume of the solid bounded by $$z = 4 - \sqrt{x^2+y^2}$$, $$x^2 + y^2 = 1$$ and $$z = 0$$.

Problem Statement

Calculate the volume of the solid bounded by $$z = 4 - \sqrt{x^2+y^2}$$, $$x^2 + y^2 = 1$$ and $$z = 0$$.

$$10 \pi/3$$

Problem Statement

Calculate the volume of the solid bounded by $$z = 4 - \sqrt{x^2+y^2}$$, $$x^2 + y^2 = 1$$ and $$z = 0$$.

Solution

### 1971 video

video by MIP4U

$$10 \pi/3$$

Evaluate $$\iiint_{V}{ x^2 ~dV }$$ where V is the solid that lies within the cylinder $$x^2 + y^2 = 1$$ above the plane $$z=0$$ and below the cone $$z^2 = 4x^2 + 4y^2$$.

Problem Statement

Evaluate $$\iiint_{V}{ x^2 ~dV }$$ where V is the solid that lies within the cylinder $$x^2 + y^2 = 1$$ above the plane $$z=0$$ and below the cone $$z^2 = 4x^2 + 4y^2$$.

$$2\pi/5$$

Problem Statement

Evaluate $$\iiint_{V}{ x^2 ~dV }$$ where V is the solid that lies within the cylinder $$x^2 + y^2 = 1$$ above the plane $$z=0$$ and below the cone $$z^2 = 4x^2 + 4y^2$$.

Solution

Note: Although she calls the result of this integral a volume, it is not truly a volume. If we were calculating the volume, then we would be integrating 1 over the solid region, not $$x^2$$.

### 1972 video

video by Krista King Math

$$2\pi/5$$

$$\displaystyle{ \int_{-1}^{1}{ \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}{ \int_{0}^{2}{ \sqrt{x^2+y^2} } } } }$$ $$dz ~dy ~dx$$

Problem Statement

$$\displaystyle{ \int_{-1}^{1}{ \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}{ \int_{0}^{2}{ \sqrt{x^2+y^2} } } } }$$ $$dz ~dy ~dx$$

$$4 \pi/3$$

Problem Statement

$$\displaystyle{ \int_{-1}^{1}{ \int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}{ \int_{0}^{2}{ \sqrt{x^2+y^2} } } } }$$ $$dz ~dy ~dx$$

Solution

### 1973 video

video by MIP4U

$$4 \pi/3$$

$$\displaystyle{ \int_{-2}^{2}{ \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}{ \int_{x^2+y^2}^{4}{ y~dz~dy~dx } } } }$$

Problem Statement

$$\displaystyle{ \int_{-2}^{2}{ \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}{ \int_{x^2+y^2}^{4}{ y~dz~dy~dx } } } }$$

0

Problem Statement

$$\displaystyle{ \int_{-2}^{2}{ \int_{-\sqrt{4-x^2}}^{\sqrt{4-x^2}}{ \int_{x^2+y^2}^{4}{ y~dz~dy~dx } } } }$$

Solution

### 1974 video

video by MIP4U

0

Evaluate $$\iiint\limits_E e^z ~dV$$ where E is enclosed by the paraboloid $$z = 3 + x^2 + y^2$$, the cylinder $$x^2 + y^2 = 2$$ and the xy-plane.

Problem Statement

Evaluate $$\iiint\limits_E e^z ~dV$$ where E is enclosed by the paraboloid $$z = 3 + x^2 + y^2$$, the cylinder $$x^2 + y^2 = 2$$ and the xy-plane.

Solution

### 2307 video

video by MIP4U

Intermediate Problems

Integrate $$f(x,y,z) = 1$$ over the solid region bounded by $$z = 0$$, $$x^2 + y^2 = 1$$, $$x^2 + y^2 = 4$$ and $$z = 4 - ( x^2 + y^2 )$$ in cylindrical coordinates.

Problem Statement

Integrate $$f(x,y,z) = 1$$ over the solid region bounded by $$z = 0$$, $$x^2 + y^2 = 1$$, $$x^2 + y^2 = 4$$ and $$z = 4 - ( x^2 + y^2 )$$ in cylindrical coordinates.

$$9 \pi/2$$

Problem Statement

Integrate $$f(x,y,z) = 1$$ over the solid region bounded by $$z = 0$$, $$x^2 + y^2 = 1$$, $$x^2 + y^2 = 4$$ and $$z = 4 - ( x^2 + y^2 )$$ in cylindrical coordinates.

Solution

### 1930 video

video by MIP4U

$$9 \pi/2$$

Express as a triple integral, using cylindrical coordinates, the volume of the region above the cone $$z = \sqrt{x^2+y^2}$$ and inside the sphere $$x^2 + y^2 + z^2 = 2az, a > 0$$.

Problem Statement

Express as a triple integral, using cylindrical coordinates, the volume of the region above the cone $$z = \sqrt{x^2+y^2}$$ and inside the sphere $$x^2 + y^2 + z^2 = 2az, a > 0$$.

$$\displaystyle{ \int_{0}^{2\pi}{ \int_{0}^{a}{ \int_{r}^{a+\sqrt{a^2-r^2}}{ } } } }$$ $$r~dz~dr~d\theta$$

Problem Statement

Express as a triple integral, using cylindrical coordinates, the volume of the region above the cone $$z = \sqrt{x^2+y^2}$$ and inside the sphere $$x^2 + y^2 + z^2 = 2az, a > 0$$.

Solution

Note - This same problem is set up in rectangular coordinates on the rectangular coordinates page and in spherical coordinates on the spherical coordinates page. Also on the spherical coordinates page, the integral is evaluated. But you are not asked to evaluate it here, in rectangular coordinates.

### 1932 video

video by Dr Chris Tisdell

$$\displaystyle{ \int_{0}^{2\pi}{ \int_{0}^{a}{ \int_{r}^{a+\sqrt{a^2-r^2}}{ } } } }$$ $$r~dz~dr~d\theta$$

Use cylindrical coordinates to find the volume outside the cylinder $$x^2 + y^2 = 1$$, ($$z \ge 0$$) and inside the paraboloid $$z = 4 - x^2 - y^2$$.

Problem Statement

Use cylindrical coordinates to find the volume outside the cylinder $$x^2 + y^2 = 1$$, ($$z \ge 0$$) and inside the paraboloid $$z = 4 - x^2 - y^2$$.

$$9 \pi/2$$

Problem Statement

Use cylindrical coordinates to find the volume outside the cylinder $$x^2 + y^2 = 1$$, ($$z \ge 0$$) and inside the paraboloid $$z = 4 - x^2 - y^2$$.

Solution

### 1934 video

video by Dr Chris Tisdell

$$9 \pi/2$$

Find the volume between the paraboloid $$z = x^2 + y^2$$ and the plane $$z = 2y$$.

Problem Statement

Find the volume between the paraboloid $$z = x^2 + y^2$$ and the plane $$z = 2y$$.

$$\pi/2$$

Problem Statement

Find the volume between the paraboloid $$z = x^2 + y^2$$ and the plane $$z = 2y$$.

Solution

### 1969 video

video by MIT OCW

$$\pi/2$$