Limits Derivatives Integrals Infinite Series Parametrics Polar Coordinates Conics
Limits
Epsilon-Delta Definition
Finite Limits
One-Sided Limits
Infinite Limits
Trig Limits
Pinching Theorem
Indeterminate Forms
L'Hopitals Rule
Limits That Do Not Exist
Continuity & Discontinuities
Intermediate Value Theorem
Derivatives
Power Rule
Product Rule
Quotient Rule
Chain Rule
Trig and Inverse Trig
Implicit Differentiation
Exponentials & Logarithms
Logarithmic Differentiation
Hyperbolic Functions
Higher Order Derivatives
Differentials
Slope, Tangent, Normal...
Linear Motion
Mean Value Theorem
Graphing
1st Deriv, Critical Points
2nd Deriv, Inflection Points
Related Rates Basics
Related Rates Areas
Related Rates Distances
Related Rates Volumes
Optimization
Integrals
Definite Integrals
Integration by Substitution
Integration By Parts
Partial Fractions
Improper Integrals
Basic Trig Integration
Sine/Cosine Integration
Secant/Tangent Integration
Trig Integration Practice
Trig Substitution
Linear Motion
Area Under/Between Curves
Volume of Revolution
Arc Length
Surface Area
Work
Moments, Center of Mass
Exponential Growth/Decay
Laplace Transforms
Describing Plane Regions
Infinite Series
Divergence (nth-Term) Test
p-Series
Geometric Series
Alternating Series
Telescoping Series
Ratio Test
Limit Comparison Test
Direct Comparison Test
Integral Test
Root Test
Absolute Convergence
Conditional Convergence
Power Series
Taylor/Maclaurin Series
Radius of Convergence
Interval of Convergence
Remainder & Error Bounds
Fourier Series
Study Techniques
Choosing A Test
Sequences
Infinite Series Table
Practice Problems
Exam Preparation
Exam List
Parametrics
Parametric Curves
Parametric Surfaces
Slope & Tangent Lines
Area
Arc Length
Surface Area
Volume
Polar Coordinates
Converting
Slope & Tangent Lines
Area
Arc Length
Surface Area
Conics
Parabolas
Ellipses
Hyperbolas
Conics in Polar Form
Vectors Vector Functions Partial Derivatives/Integrals Vector Fields Laplace Transforms Tools
Vectors
Unit Vectors
Dot Product
Cross Product
Lines In 3-Space
Planes In 3-Space
Lines & Planes Applications
Angle Between Vectors
Direction Cosines/Angles
Vector Projections
Work
Triple Scalar Product
Triple Vector Product
Vector Functions
Projectile Motion
Unit Tangent Vector
Principal Unit Normal Vector
Acceleration Vector
Arc Length
Arc Length Parameter
Curvature
Vector Functions Equations
MVC Practice Exam A1
Partial Derivatives
Gradients
Directional Derivatives
Lagrange Multipliers
Tangent Plane
MVC Practice Exam A2
Partial Integrals
Describing Plane Regions
Double Integrals-Rectangular
Double Integrals-Applications
Double Integrals-Polar
Triple Integrals-Rectangular
Triple Integrals-Cylindrical
Triple Integrals-Spherical
MVC Practice Exam A3
Vector Fields
Curl
Divergence
Conservative Vector Fields
Potential Functions
Parametric Curves
Line Integrals
Green's Theorem
Parametric Surfaces
Surface Integrals
Stokes' Theorem
Divergence Theorem
MVC Practice Exam A4
Laplace Transforms
Unit Step Function
Unit Impulse Function
Square Wave
Shifting Theorems
Solve Initial Value Problems
Prepare For Calculus 1
Ready For Calculus 2?
Trig Formulas
Describing Plane Regions
Parametric Curves
Linear Algebra Review
Word Problems
Mathematical Logic
Calculus Notation
Simplifying
Practice Exams
17calculus on YouTube
More Math Help
Tutoring
Tools and Resources
Academic Integrity
Learning/Study Techniques
Math/Science Learning
Memorize To Learn
Music and Learning
Note-Taking
Motivation
Instructor or Coach?
Books
Math Books
How To Read Math Books

You CAN Ace Calculus

17calculus > parametrics > calculus

### Calculus Main Topics

Single Variable Calculus

Multi-Variable Calculus

### Tools

math tools

general learning tools

additional tools

### Related Topics and Links

Parametric Equations and Calculus

On this page we cover the most common calculus problems using parametric equations. The two main topics are differentiation and integration.

### Search 17Calculus

Loading

This table lists the topics on this page with links to jump to the topic you are interested in.

 Differentiation Integration Slope and Tangent Lines Area Surface Area Higher Order Derivatives Arc Length Volume

Let's start out with a quick video clip giving us an introduction to finding the derivative $$dy/dx$$ if the function $$y=f(x)$$ is given in parametric equations $$x(t)$$ and $$y(t)$$. The theorem is given below.

 PatrickJMT - Parametric Differentiation

Theorem: Parametric Derivative

On a smooth curve given by the equations $$x=X(t)$$ and $$y = Y(t)$$,
the slope of the curve at the point $$(x,y)$$ is $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} \text{ where } dx/dt \neq 0$

### Parametric Derivative Proof

Theorem: Parametric Derivative

On a smooth curve given by the equations $$x=X(t)$$ and $$y = Y(t)$$, the slope of the curve at the point $$(x,y)$$ is $\frac{dy}{dx} = \frac{dy/dt}{dx/dt} \text{ where } dx/dt \neq 0$

We present two proofs here, a short informal version and a longer, more formal version.

Short Informal Proof
Given $$y(t)$$ and $$x(t)$$, we can write $$y = y(x(t))$$. We want to find $$dy/dx$$.
Using the chain rule on $$y(x(t))$$, we have $$dy/dt = dy/dx \cdot dx/dt$$.
Solving for $$dy/dx$$ we have $$\displaystyle{ dy/dx = \frac{dy/dt}{dx/dt} }$$      [qed]

Longer More Formal Proof
Given the two points $$(x_1,y_1) = (X(t), Y(t))$$ and $$(x_2,y_2) = (X(t+\Delta t), Y(t+\Delta t))$$ and $$\Delta t > 0$$ on a smooth curve, let $$\Delta x = x_2 - x_1 = X(t+\Delta t) - X(t)$$ and $$\Delta y = y_2 - y_1 = Y(t+\Delta t) - Y(t)$$.

As $$\Delta t \to 0$$, we know that $$\Delta x \to 0$$ and we can write

$$\displaystyle{ \frac{dy}{dx} = \lim_{\Delta x \to 0}{\frac{\Delta y}{\Delta x}} = \lim_{\Delta t \to 0}{\frac{Y(t+\Delta t) - Y(t)}{X(t+\Delta t) - X(t)}} }$$

Now we can multiply the numerator and denominator by $$1/\Delta t$$ and use limit laws to give us

$$\begin{array}{rcl} \displaystyle{\frac{dy}{dx}} & = & \displaystyle{\lim_{\Delta t \to 0}{\frac{[Y(t+\Delta t) - Y(t)](1/\Delta t)}{[X(t+\Delta t) - X(t)](1/\Delta t)}} } \\ & = & \displaystyle{ \frac{\lim_{\Delta t \to 0}{[Y(t+\Delta t) - Y(t)]/\Delta t}}{\lim_{\Delta t \to 0}{[X(t+\Delta t) - X(t)]/\Delta t}} } \\ & = & \displaystyle{ \frac{dy/dt}{dx/dt} ~~~~~ \text{ [qed] } } \end{array}$$

Before we go on, let's practice using this theorem.

Practice 1

Find $$dy/dx$$ of the parametric curve $$x=t+5\cos(t)$$, $$y=3e^t$$.

solution

Practice 2

Find the derivative of the parametric curve $$x=t\sin(t)$$, $$y=t^2+t$$.

solution

Practice 3

Find $$dy/dx$$ of the parametric curve $$x=4t+1$$, $$y=t^2+2t$$.

solution

Differentiation - Slope and Tangent Lines

To find the equation of a tangent line to a graph given by a set of parametric equations, we need to be able to find the slope by calculating the derivative $$dy/dx$$ using the above theorem.

For horizontal tangent lines, the slope $$dy/dx$$ is zero, so we need $$dy/dt = 0$$ and $$dx/dt \neq 0$$. For vertical tangent lines, the slope is undefined, which means that $$dx/dt = 0$$ when $$dy/dt \neq 0$$. In the case where both $$dx/dt = 0$$ and $$dy/dt = 0$$ at the same point, we need to handle that case separately, since nothing can be concluded from $$dy/dx = 0/0$$, which is indeterminate.

Once you have found the slope, you can easily find the equation of a tangent line. Go to the Tangent Lines page for more information.

Practice 4

Find the equation of the tangent line to the parametric curve $$x=2t^2+1$$, $$y=3t^3+2$$ at $$t=1$$.

solution

Practice 5

Find the equation of the tangent line to the parametric curve $$x=3(t-\sin(t))$$, $$y=3(1-\cos(t))$$ at $$t=\pi/2$$.

solution

Practice 6

Find the equation of the tangent line to the parametric curve $$x=t\cos(t)$$, $$y=t\sin(t)$$ at $$t=\pi$$.

solution

Practice 7

Find the equation of the tangent line to the parametric curve $$x=3t^2-t$$, $$y=\sqrt{t}$$ at $$t=4$$.

solution

Differentiation - Higher Order Derivatives

In order to determine concavity of a graph and other information, you will need higher order derivatives. We list two of them below from which you can extract a pattern.

Second Derivative

$$\displaystyle{ \frac{d^2y}{dx^2} = \frac{d}{dx}\left[ \frac{dy}{dx} \right] = \frac{d\left[ dy/dx \right]/dt}{dx/dt} }$$
You may also find the notation $$\dot{x} = dx/dt$$, in which case the above derivative can be written
$$\displaystyle{ \frac{d^2y}{dx^2} = \frac{\dot{x} \ddot{y} - \dot{y} \ddot{x}}{\dot{x}^3} }$$

Third Derivative

$$\displaystyle{ \frac{d^3y}{dx^3} = \frac{d}{dx}\left[ \frac{d^2y}{dx^2} \right] = \frac{d\left[ d^2y/dx^2 \right] /dt}{dx/dt} }$$

Before we go on, let's work some practice problems with what we have learned so far.

 Basic Problems

Practice 8

Find the second derivative of the parametric equations $$x=t^2+t$$, $$y=2t-1$$.

solution

 Intermediate Problems

Practice 9

Find the second derivative of the parametric curve $$x=t^3+t$$, $$y=t^5+1$$.

solution

Practice 10

Find the second derivative of the parametric curve $$x=t-t^3$$, $$y= 2t+5$$.

solution

Practice 11

Find the horizontal tangent points to the curve $$x=1-t, y=t^2$$ and determine the concavity at those points.

answer

solution

Practice 12

For the parametric curve $$x=\sqrt{t}, y=3t-1$$, find the derivative and the slope at the point $$t=1$$. Also determine the concavity and the equation of the tangent line at that same point. Give your equation of the tangent line in slope-intercept form.

answer

solution

Integration - Area

The area under a smooth curve defined parametrically as $$x = X(t)$$ and $$y = Y(t)$$ from $$t=t_0$$ to $$t=t_1$$ can be calculated using the integral

$$\displaystyle{s = \int_{t_0}^{t_1}{Y(t)X'(t) ~ dt}}$$     where $$X'(t) = dX/dt$$.

You may also find this written in a shorthand form as

$$\displaystyle{s = \int_{t_0}^{t_1}{y~dx}}$$

In this notation, $$dx = (dx/dt) dt$$.

Practice 13

Find the area under the curve $$x=1+e^t$$, $$y=t-t^2$$ and above the x-axis.

solution

Practice 14

Calculate the area enclosed by the line $$y=2.5$$ and the parametric curve $$x=t-1/t$$, $$y=t+1/t$$.

answer

solution

Practice 15

Calculate the area under one arc of the parametric curve $$x=r\theta-d\sin\theta$$, $$y=r-d\cos\theta$$.

solution

Integration - Arc Length

When a smooth curve is defined parametrically as $$x=X(t)$$ and $$y = Y(t)$$, the arc length between the points $$t=t_0$$ and $$t = t_1$$ can be calculated using the integral

$$\displaystyle{ s = \int_{t_0}^{t_1}{\sqrt{[X'(t)]^2 + [Y'(t)]^2} dt}}$$
Notice we use a small s here to represent the arc length. This is the standard symbol you will see in many textbooks. We reserve a capital S to represent surface area.

Here is a quick video clip going over this equation in more detail.

 PatrickJMT - Arc Length
 Basic Problems

Practice 16

Compute the arc length of the parametric curve $$x(t)=3t^2-9$$, $$y(t)=t^3-3t$$, $$0\leq t\leq 3$$.

answer

solution

Practice 17

Calculate the arc length of the parametric curve $$x(t)=t\sin(t)$$ $$y(t)=t\cos(t)$$, $$0\leq t\leq1$$.

answer

solution

Practice 18

Calculate the arc length of the parametric curve $$x=1+3t^2$$, $$y=4+2t^3$$, $$0\leq t\leq1$$.

answer

solution

Practice 19

Calculate the arc length of the parametric curve $$x=\abs{6-t}$$, $$y=t$$, $$0 \leq t \leq 3$$.

answer

solution

 Intermediate Problems

Practice 20

Find the arc length of the parametric curve $$x=2t$$, $$y=(2/3)t^{3/2}$$ for $$5\leq t\leq12$$.

answer

solution

Practice 21

Calculate the arc length of the parametric curve $$x=t^3$$, $$y=t^2$$ from $$(0,0)$$ to $$(8,4)$$.

answer

solution

Practice 22

Calculate the arc length of the curve $$x(\theta)=a\cos^3(\theta)$$, $$y(\theta)=a\sin^3(\theta)$$, $$0\leq\theta\leq2\pi$$.

answer

solution

Practice 23

Calculate the arc length of the parametric curve $$x=e^t+e^{-t}$$, $$y=5-2t$$, $$0\leq t\leq3$$.

answer

solution

Practice 24

Calculate the arc length of the parametric curve $$x=e^t\cos(t)$$, $$y=e^t\sin(t)$$, $$0\leq t\leq\pi$$.

answer

solution

Integration - Surface Area

To calculate the surface area defined by revolving a parametric curve defined as $$x=X(t)$$ and $$y=Y(t)$$ from $$t=t_0$$ to $$t=t_1$$ about the x-axis, we use this integral.

$$\displaystyle{S = 2\pi ~ \int_{t_0}^{t_1}{ Y(t) \sqrt{[X'(t)]^2 + [Y'(t)]^2} ~ dt}}$$

If you compare this integral to the equation for arc length (in the prevous section) you will see the common factor $$\sqrt{[X'(t)]^2 + [Y'(t)]^2}$$

Some textbooks write the surface integral differently, taking this into account. You may see it written as

$$\displaystyle{S = 2\pi ~ \int_{t_0}^{t_1}{Y(t) ~ ds}}$$
where $$ds = \sqrt{[X'(t)]^2 + [Y'(t)]^2} ~dt$$

When rotating about the y-axis, the integral we use is

$$\displaystyle{S = 2\pi ~ \int_{t_0}^{t_1}{X(t) ~ ds}}$$

Notice we use a capital S to represent surface area. This is the standard symbol you will see in many textbooks. We reserve a lowercase s to represent arc length.

Practice 25

Find the surface area of revolution of the parametric curve $$x=3t^2$$, $$y=2t^3$$, $$0\leq t\leq5$$ rotated about the y-axis.

answer

solution

Practice 26

Find the surface area of revolution of the parametric curve $$x=1-t$$, $$y=2\sqrt{t}$$, $$1 \leq t \leq 4$$ rotated about the x-axis.

answer

solution

Integration - Volume

If we have a parametric curve defined as $$x = X(t)$$ and $$y = Y(t)$$, we can determine the volume of the solid object defined by revolving this curve about an axis. We will limit our curve from $$t=t_0$$ to $$t=t_1$$.

Revolution About the x-axis Revolution About the y-axis When revolved about the x-axis, the integral is $$\displaystyle{V_x = \pi \int_{t_0}^{t_1}{y^2 [dx/dt] dt}}$$ This is sometimes written as $$\displaystyle{V_x = \pi \int_{t_0}^{t_1}{y^2 ~ dx}}$$ where $$dx = [dx/dt] dt$$ Similar to the x-axis integral, we have $$\displaystyle{V_y = \pi \int_{t_0}^{t_1}{x^2 [dy/dt] dt} = \pi \int_{t_0}^{t_1}{x^2 ~ dy}}$$ where $$dy = [dy/dt] dt$$

Practice 27

Calculate the volume of revolution when the curve $$x=t^3$$, $$y=2t^2+1$$, $$-1\leq t \leq 1$$ is rotated about the x-axis.

answer

solution

8