Area of parametric equations - Parametric Equations and Polar Coordinates

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=aby  dx=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=aby  dx=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!
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Area of parametric equations

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