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)x=f(t), y=g(t)y=g(t) and let the value of tt be increasing from α\alpha to β\beta. Then we say that the formula for the length of the curve is:
L=αβ(dxdt)2+(dydt)2dtL=\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 xx-axis, where f,gf', g' are continuous and g(t)0g(t) \geq 0, then the formula for the surface area of the curve is
SA=αβ2πy(dxdt)2+(dydt)2dtSA=\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=etsint x=e^t \sin t
    y=etcosty=e^t \cos t
    where 0t2π0 \leq t \leq 2\pi

    b)
    x=cos(θ) x=\cos (\theta)
    y=sin(θ)y=\sin (\theta)
    where 0θπ0 \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 xx-axis:
    a)
    x=4tt2x=4t-t^2
    y=2ty=2t
    where 0t30 \leq t \leq 3

    b)
    x=r(θsinθ)x=r(\theta - \sin \theta)
    y=r(1cosθ)y=r(1- \cos \theta ) where 0t2π0 \leq t \leq 2\pi


  • 3.
    Applications related to Circles and Spheres
    You are given the parametric equations x=rcos(t)x=r\; \cos(t), y=rsin(t)y=r\;\sin(t) where 0t2π0 \leq t \leq 2\pi. Show that the circumference of a circle is 2πr2\pi r

  • 4.
    You are given the parametric equations x=rcos(t)x=r\; \cos(t), y=rsin(t)y=r\;\sin(t) where 0tπ0 \leq t \leq \pi. Show that the surface area of a sphere is 4πr24\pi r^2