# Arc length and surface area of parametric equations

### Arc length and surface area of parametric equations

In this lesson, we will learn how to find the arc length and surface area of parametric equations. To find the arc length, we have to integrate the square root of the sums of the squares of the derivatives. For surface area, it is actually very similar. If it is rotated around the x-axis, then all you have to do is add a few extra terms to the integral. Note that integrating these are very hard, and would require tons of trigonometric identity substitutions to make it simpler. We will first apply these formulas to some of the questions below. Then we will look at a case where using these formulas will give us much more simplified formulas in finding the arc length and surface areas of circles and spheres.

#### Lessons

Let the curve be defined by the parametric equations $x=f(t)$, $y=g(t)$ and let the value of $t$ be increasing from $\alpha$ to $\beta$. Then we say that the formula for the length of the curve is:
$L=\int_{\alpha}^{\beta}\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}dt$

The formula to find the surface area is very similar.

If the curve is rotating around the $x$-axis, where $f', g'$ are continuous and $g(t) \geq 0$, then the formula for the surface area of the curve is
$SA=\int_{\alpha}^{\beta} 2\pi y\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}dt$
• Introduction
Overview:
a)
Arc Length of Parametric Equations

b)
Surface Area of Parametric Equations

• 1.
The Length of a Curve
Find the length of each of the given parametric equations:
a)
$x=e^t \sin t$
$y=e^t \cos t$
where $0 \leq t \leq 2\pi$

b)
$x=\cos (\theta)$
$y=\sin (\theta)$
where $0 \leq \theta \leq \pi$

• 2.
The Surface Area of a Curve rotating about the x-axis
Find the surface area for each of the given parametric equations by rotating about the $x$-axis:
a)
$x=4t-t^2$
$y=2t$
where $0 \leq t \leq 3$

b)
$x=r(\theta - \sin \theta)$
$y=r(1- \cos \theta )$ where $0 \leq \theta \leq 2\pi , \; r > 0$

• 3.
Applications related to Circles and Spheres
You are given the parametric equations $x=r\; \cos(t)$, $y=r\;\sin(t)$ where $0 \leq t \leq 2\pi$. Show that the circumference of a circle is $2\pi r$

• 4.
You are given the parametric equations $x=r\; \cos(t)$, $y=r\;\sin(t)$ where $0 \leq t \leq \pi$. Show that the surface area of a sphere is $4\pi r^2$