In this video I go over further into Calculus with Parametric Curves and this time look at calculating the area underneath a parametric curve. The area underneath a curve y = F(x) can be calculated by using definite integrals. But for parametric equations x = f(t) and y = g(t), I show that we can use the Substitution Rule for Definite integrals to replace the x term with the parameter t. But when we do this it is important to ensure that the interval of the parameter is such that the curve is traced just once, otherwise we will obtain a larger area. This is a very useful video on using the substitution rule to re-arrange definite integrals to account for parametric equations, and which I will be using in later videos, so make sure to watch this video!
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Areas
We know that the area under a curve y = F(x) from a to b:
If the curve is given by parametric equations x = f(t), and y = g(t) and is traversed once as t increases from α to β, then we can adapt the earlier formula by using the Substitution Rule for Definite Integrals as follows:
