Area of parametric equations

Area of parametric equations

In this section, we will learn find the area under the curve of parametric equations. This still involves integration, but the integrand looks changed. The integrand is now the product between the second function and the derivative of the first function. We will examine the different types of parametric equations with a given range, and learn how to find the area of each one.


Normally we know that the area under the curve from aa to bb is abf(x)dx\int_{a}^{b} f(x)dx. However, what about parametric equations?
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 area under the parametric curve is:

A=abydx=αβg(t)f(t)dtA = \int_{a}^{b} y \; dx=\int_{\alpha}^{\beta} g(t)f'(t)dt

However, if the value of tt is increasing from β\beta to α\alpha instead, then the area under the parametric curve will be:

A=abydx=βαg(t)f(t)dtA = \int_{a}^{b} y \; dx=\int_{\beta}^{\alpha} g(t)f'(t)dt

Be careful when determining which one to use!
  • Introduction
    Area of Parametric Functions Overview

  • 1.
    Finding the Area Given the Range of the Parameter
    Find the area under the curve of the parametric curve x=t2+1x=t^2+1
    y=t3+t2+4y=t^3+t^2+4, where 1t31 \leq t \leq 3.
    Assume that the curve traces perfectly from left to right for the range of the parameter tt.

  • 2.
    Find the area enclosed of the given parametric curve x=acos(θ)x=a \cos (\theta), y=bsin(θ)y= b \sin (\theta), where 0θ2π0 \leq \theta \leq 2 \pi and a,ba, b are constants.

  • 3.
    Find the area under the curve of the parametric equations x=t1tx=t-\frac{1}{t}, y=t+1ty=t+\frac{1}{t}, where 12t2 \frac{1}{2} \leq t \leq 2 .