can we find the area under a curve that is not a function?

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I am doing parametric calculus now, and I went the process of finding the area under a parametric curve, and one of the conditions to find the is that the curve has to represent a function on that interval. My question is that cant we find the area under a curve that is not a function, but we can break it or cut it to into subintervals so that these subintervals do represent a function and find the areas of these subintervals and then sum them up?