Area with Parametric Equations

Area with Parametric Equations - Area with Parametric...

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Area with Parametric Equations In this section we will find a formula for determining the area under a parametric curve given by the parametric equations, We will also need to further add in the assumption that the curve is traced out exactly once as t increases from α to β. We will do this in much the same way that we found the first derivative in the previous section. We will first recall how to find the area under on . We will now think of the parametric equation as a substitution in the integral. We will also assume that and for the purposes of this formula. There is actually no reason to assume that this will always be the case and so we’ll give a corresponding formula later if it’s the opposite case ( and ). So, if this is going to be a substitution we’ll need, Plugging this into the area formula above and making sure to change the limits to their corresponding t values gives us,
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Since we don’t know what F(x) is we’ll use the fact that and we arrive at the formula that we want.
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