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9.3 Finding Arc Lengths of Curves Given by Parametric Equations

5 min readjanuary 24, 2023

For this portion of the AP Calculus BC course, we will learn how to apply what we have previously learned about arc length to parametric curves. To learn more about finding arc lengths of smooth planar curves, please refer to the Unit 8.13 study guide!

🏹 Reviewing Arc Length

In calculus, the arc length of a curve refers to the distance between two points on a curve. For example, if we were to mark two points on a paperclip, we could measure the arc length between those two points by unraveling the paperclip. Once the paperclip is straightened out, we can use a ruler to measure the distance between those two points. A similar strategy is used to measure the arc length of a curve.

Untitled

Image Courtesy of Math Is Fun

✏️ Derivation of Arc Length Formula (Cartesian)

This image above depicts how calculus can be used to calculate the arc length of curves. The process essentially involves using the Pythagorean Theorem, c=a2+b2c=\sqrt{a^2+b^2}, to find the hypotenuse of a triangle with side lengths of dxdx and dydy. By adding up all the little hypotenuses, we can get a good approximation for the arc length of the curve. The arc length formula is derived from this idea. The derivation is shown in greater detail below:

ds2=dxdx2+dydx2ds^2 = \tfrac{dx}{dx}^2 + \tfrac{dy}{dx}^2
ds=dxdx2+dydx2ds=\sqrt{\tfrac{dx}{dx}^2 + \tfrac{dy}{dx}^2}
ds=1+dydx2ds=\sqrt{1+\tfrac{dy}{dx}^2}
S=ab1+[f(x)]2dxS=\int_a^b \sqrt{1+[f'(x)]^2} dx

However, this formula is only used to find the arc length of smooth planar curves. We need to adapt the formula a little more in order to use it for parametric curves!


💭 Arc Length of Parametric Curves

Review of Parametric Equations

To understand how the arc length formula needs to be adapted for parametric equations, it’s important that you first understand the relationship between parametric and Cartesian equations. The diagram below does a great job of helping us visualize this relationship!

Untitled

Image Courtesy of GeoGebra

As can be seen in the image, x(t)=rcos(θ)x(t)= r cos(θ) while y(t)=rsin(θ)y(t)= r sin(θ) .

✏️ Deriving the Arc Length Formula (Parametric)

Earlier, we derived the arc length formula for cartesian curves. When we substitute dx/dxdx/dx for dy/dxdy/dx for their parametric equivalents, dx(t)dt\tfrac{dx(t)}{dt} and dy(t)dt\tfrac{dy(t)}{dt}, we arrive at the arc length formula for parametric curves. This process is shown below:

ds2=dx(t)dt2+dy(t)dt2ds^2 = \tfrac{dx(t)}{dt}^2 + \tfrac{dy(t)}{dt}^2
ds=dx(t)dt2+dy(t)dt2ds=\sqrt{\tfrac{dx(t)}{dt}^2 + \tfrac{dy(t)}{dt}^2}
S=ab(dx(t)dt)2+(dy(t)dt)2dt\boxed{S=\int_a^b \sqrt{(\tfrac{dx(t)}{dt})^2 + (\tfrac{dy(t)}{dt})^2} dt }

Notice that the formula looks similar to the case of the Cartesian equation. However, both x and y are changing, so we need to account for both of their fluctuations!


📝 Arc Length—Parametric Practice

Now that we know the formula for finding the arc length of parametric curves, let’s try a couple of practice problems!

Arc Length Practice Question 1

Find the arc length over [0, π] for the parametric curve x(t)=sin(t)x(t) = sin(t) and y(t)=cos(t)y(t)=cos(t).

Here’s how we can go about this question. First, we can begin by finding taking the derivatives of x(t)x(t) and y(t)y(t).

dx(t)dt=cos(t)\tfrac{dx(t)}{dt} = cos(t)
dy(t)dt=sin(t)\tfrac{dy(t)}{dt} = - sin(t)

Recall that the formula for arc length is S=ab(dx(t)dt)2+(dy(t)dt)2dtS=\int_a^b \sqrt{(\tfrac{dx(t)}{dt})^2 + (\tfrac{dy(t)}{dt})^2} dt.

We are given the values of aa and bb in the problem. We know that a=0a=0 and b=πb=π. All we have left to do is substitute our values into the problem and solve it.

S=0π(cos(t))2+(sin(t))2dtS=\int_0^π \sqrt{(cos(t))^2 + (-sin(t))^2} dt
S=0π(1)dtS=\int_0^π \sqrt{(1)} dt

We can simplify (cos(t))2+(sin(t))2(cos(t))^2 + (-sin(t))^2 to cos2(t)+sin2(t)cos^2(t) + sin^2(t) which is equivalent to 1!

S=π0S=π -0
S=πS=π

And we have our answer! The arc length is ππ.

Arc Length Practice Question 2

Let’s try solving another problem!

Find the arc length over [0, π] for the parametric curve x(t)=2x(t) = 2 and y(t)=t2y(t)=t^2.

Once again, we can begin by finding taking the derivatives of x(t)x(t) and y(t)y(t).

dx(t)dt=0\tfrac{dx(t)}{dt} = 0

dy(t)dt=2t\tfrac{dy(t)}{dt} = 2t

The formula for arc length is S=ab(dx(t)dt)2+(dy(t)dt)2dtS=\int_a^b \sqrt{(\tfrac{dx(t)}{dt})^2 + (\tfrac{dy(t)}{dt})^2} dt.

Like last time, we are given the values of aa and bb in the problem. We know that a=0a=0 and b=πb=π.

Now we have to substitute our values into the problem and solve it.

S=0π(0)2+(2t)2dtS=\int_0^π \sqrt{(0)^2 + (2t)^2} dt
S=0π(4t2)dtS=\int_0^π \sqrt{(4t^2)} dt
S=0π2tdtS=\int_0^π 2t dt
S=[4t2]0πS= [4t^2]_0^π
S=4π2S= 4 π^2

And we have our answer! The arc length is 4π24π^2 .


⭐ Closing

These problems rarely get tougher than this!! The difficult part is integrating the equations. Once you get the hang of integrating, you will be easily able to solve these problems :)

Key Terms to Review (13)

Accumulation of Distance

: The accumulation of distance refers to the total distance traveled by an object over a given time interval. It takes into account both positive and negative distances.

Arc Length

: The arc length of a curve is the distance along the curve between two points. It measures how long the curve is.

Cartesian Equations

: Cartesian equations are algebraic equations that describe curves or shapes on a coordinate plane using x and y variables. They are named after René Descartes, who introduced this method of representing geometric figures.

Definite Integral

: A definite integral is a mathematical tool used to calculate the exact area between a function and the x-axis over a specific interval.

Integrals

: Integrals are mathematical tools used to find the area under a curve or to calculate the accumulation of quantities over a given interval.

Limits of Integration

: The limits of integration are the values that determine the range over which an integral is evaluated. They specify the starting and ending points on the x-axis for finding the area or calculating other quantities using integration.

Numerical Methods

: Numerical methods are techniques used to approximate solutions to mathematical problems when exact solutions are difficult or impossible to find analytically. These methods involve using algorithms, computations, and iterative processes to obtain numerical approximations.

Parametric Equations

: Parametric equations are a set of equations that express the coordinates of points on a curve or surface in terms of one or more parameters. They allow us to represent complex shapes and motions by breaking them down into simpler components.

Parametric Functions

: Parametric functions are a way to represent curves or graphs using two separate equations, one for the x-coordinate and one for the y-coordinate. These equations are typically in terms of a third variable called the parameter.

Partial Derivatives

: Partial derivatives are derivatives taken with respect to one variable while holding other variables constant. They measure how much a function changes when only one variable changes at a time.

Polar Equations

: Polar equations are mathematical expressions that describe curves in the polar coordinate system, where points are represented by their distance from the origin and their angle from a reference line.

Simpson's Rule

: Simpson's Rule is a numerical integration method used to approximate the definite integral of a function. It divides the interval into multiple subintervals and uses quadratic polynomials to estimate the area under the curve.

Velocity Vectors

: Velocity vectors represent the speed and direction of an object's motion at a specific point in time. They are typically represented as arrows, with the length of the arrow indicating the magnitude of velocity and the direction pointing towards the object's movement.

9.3 Finding Arc Lengths of Curves Given by Parametric Equations

5 min readjanuary 24, 2023

For this portion of the AP Calculus BC course, we will learn how to apply what we have previously learned about arc length to parametric curves. To learn more about finding arc lengths of smooth planar curves, please refer to the Unit 8.13 study guide!

🏹 Reviewing Arc Length

In calculus, the arc length of a curve refers to the distance between two points on a curve. For example, if we were to mark two points on a paperclip, we could measure the arc length between those two points by unraveling the paperclip. Once the paperclip is straightened out, we can use a ruler to measure the distance between those two points. A similar strategy is used to measure the arc length of a curve.

Untitled

Image Courtesy of Math Is Fun

✏️ Derivation of Arc Length Formula (Cartesian)

This image above depicts how calculus can be used to calculate the arc length of curves. The process essentially involves using the Pythagorean Theorem, c=a2+b2c=\sqrt{a^2+b^2}, to find the hypotenuse of a triangle with side lengths of dxdx and dydy. By adding up all the little hypotenuses, we can get a good approximation for the arc length of the curve. The arc length formula is derived from this idea. The derivation is shown in greater detail below:

ds2=dxdx2+dydx2ds^2 = \tfrac{dx}{dx}^2 + \tfrac{dy}{dx}^2
ds=dxdx2+dydx2ds=\sqrt{\tfrac{dx}{dx}^2 + \tfrac{dy}{dx}^2}
ds=1+dydx2ds=\sqrt{1+\tfrac{dy}{dx}^2}
S=ab1+[f(x)]2dxS=\int_a^b \sqrt{1+[f'(x)]^2} dx

However, this formula is only used to find the arc length of smooth planar curves. We need to adapt the formula a little more in order to use it for parametric curves!


💭 Arc Length of Parametric Curves

Review of Parametric Equations

To understand how the arc length formula needs to be adapted for parametric equations, it’s important that you first understand the relationship between parametric and Cartesian equations. The diagram below does a great job of helping us visualize this relationship!

Untitled

Image Courtesy of GeoGebra

As can be seen in the image, x(t)=rcos(θ)x(t)= r cos(θ) while y(t)=rsin(θ)y(t)= r sin(θ) .

✏️ Deriving the Arc Length Formula (Parametric)

Earlier, we derived the arc length formula for cartesian curves. When we substitute dx/dxdx/dx for dy/dxdy/dx for their parametric equivalents, dx(t)dt\tfrac{dx(t)}{dt} and dy(t)dt\tfrac{dy(t)}{dt}, we arrive at the arc length formula for parametric curves. This process is shown below:

ds2=dx(t)dt2+dy(t)dt2ds^2 = \tfrac{dx(t)}{dt}^2 + \tfrac{dy(t)}{dt}^2
ds=dx(t)dt2+dy(t)dt2ds=\sqrt{\tfrac{dx(t)}{dt}^2 + \tfrac{dy(t)}{dt}^2}
S=ab(dx(t)dt)2+(dy(t)dt)2dt\boxed{S=\int_a^b \sqrt{(\tfrac{dx(t)}{dt})^2 + (\tfrac{dy(t)}{dt})^2} dt }

Notice that the formula looks similar to the case of the Cartesian equation. However, both x and y are changing, so we need to account for both of their fluctuations!


📝 Arc Length—Parametric Practice

Now that we know the formula for finding the arc length of parametric curves, let’s try a couple of practice problems!

Arc Length Practice Question 1

Find the arc length over [0, π] for the parametric curve x(t)=sin(t)x(t) = sin(t) and y(t)=cos(t)y(t)=cos(t).

Here’s how we can go about this question. First, we can begin by finding taking the derivatives of x(t)x(t) and y(t)y(t).

dx(t)dt=cos(t)\tfrac{dx(t)}{dt} = cos(t)
dy(t)dt=sin(t)\tfrac{dy(t)}{dt} = - sin(t)

Recall that the formula for arc length is S=ab(dx(t)dt)2+(dy(t)dt)2dtS=\int_a^b \sqrt{(\tfrac{dx(t)}{dt})^2 + (\tfrac{dy(t)}{dt})^2} dt.

We are given the values of aa and bb in the problem. We know that a=0a=0 and b=πb=π. All we have left to do is substitute our values into the problem and solve it.

S=0π(cos(t))2+(sin(t))2dtS=\int_0^π \sqrt{(cos(t))^2 + (-sin(t))^2} dt
S=0π(1)dtS=\int_0^π \sqrt{(1)} dt

We can simplify (cos(t))2+(sin(t))2(cos(t))^2 + (-sin(t))^2 to cos2(t)+sin2(t)cos^2(t) + sin^2(t) which is equivalent to 1!

S=π0S=π -0
S=πS=π

And we have our answer! The arc length is ππ.

Arc Length Practice Question 2

Let’s try solving another problem!

Find the arc length over [0, π] for the parametric curve x(t)=2x(t) = 2 and y(t)=t2y(t)=t^2.

Once again, we can begin by finding taking the derivatives of x(t)x(t) and y(t)y(t).

dx(t)dt=0\tfrac{dx(t)}{dt} = 0

dy(t)dt=2t\tfrac{dy(t)}{dt} = 2t

The formula for arc length is S=ab(dx(t)dt)2+(dy(t)dt)2dtS=\int_a^b \sqrt{(\tfrac{dx(t)}{dt})^2 + (\tfrac{dy(t)}{dt})^2} dt.

Like last time, we are given the values of aa and bb in the problem. We know that a=0a=0 and b=πb=π.

Now we have to substitute our values into the problem and solve it.

S=0π(0)2+(2t)2dtS=\int_0^π \sqrt{(0)^2 + (2t)^2} dt
S=0π(4t2)dtS=\int_0^π \sqrt{(4t^2)} dt
S=0π2tdtS=\int_0^π 2t dt
S=[4t2]0πS= [4t^2]_0^π
S=4π2S= 4 π^2

And we have our answer! The arc length is 4π24π^2 .


⭐ Closing

These problems rarely get tougher than this!! The difficult part is integrating the equations. Once you get the hang of integrating, you will be easily able to solve these problems :)

Key Terms to Review (13)

Accumulation of Distance

: The accumulation of distance refers to the total distance traveled by an object over a given time interval. It takes into account both positive and negative distances.

Arc Length

: The arc length of a curve is the distance along the curve between two points. It measures how long the curve is.

Cartesian Equations

: Cartesian equations are algebraic equations that describe curves or shapes on a coordinate plane using x and y variables. They are named after René Descartes, who introduced this method of representing geometric figures.

Definite Integral

: A definite integral is a mathematical tool used to calculate the exact area between a function and the x-axis over a specific interval.

Integrals

: Integrals are mathematical tools used to find the area under a curve or to calculate the accumulation of quantities over a given interval.

Limits of Integration

: The limits of integration are the values that determine the range over which an integral is evaluated. They specify the starting and ending points on the x-axis for finding the area or calculating other quantities using integration.

Numerical Methods

: Numerical methods are techniques used to approximate solutions to mathematical problems when exact solutions are difficult or impossible to find analytically. These methods involve using algorithms, computations, and iterative processes to obtain numerical approximations.

Parametric Equations

: Parametric equations are a set of equations that express the coordinates of points on a curve or surface in terms of one or more parameters. They allow us to represent complex shapes and motions by breaking them down into simpler components.

Parametric Functions

: Parametric functions are a way to represent curves or graphs using two separate equations, one for the x-coordinate and one for the y-coordinate. These equations are typically in terms of a third variable called the parameter.

Partial Derivatives

: Partial derivatives are derivatives taken with respect to one variable while holding other variables constant. They measure how much a function changes when only one variable changes at a time.

Polar Equations

: Polar equations are mathematical expressions that describe curves in the polar coordinate system, where points are represented by their distance from the origin and their angle from a reference line.

Simpson's Rule

: Simpson's Rule is a numerical integration method used to approximate the definite integral of a function. It divides the interval into multiple subintervals and uses quadratic polynomials to estimate the area under the curve.

Velocity Vectors

: Velocity vectors represent the speed and direction of an object's motion at a specific point in time. They are typically represented as arrows, with the length of the arrow indicating the magnitude of velocity and the direction pointing towards the object's movement.


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AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.


© 2024 Fiveable Inc. All rights reserved.

AP® and SAT® are trademarks registered by the College Board, which is not affiliated with, and does not endorse this website.