Area Bounded by Closed Curve
The area bounded by a closed curve given by x = ƒ(t), y = g(t); t_{1} ≤ t ≤ t_{2} is
Proof: Let AP_{1} BP_{2} be the given closed curve which does not intersect itself. Let the curve be cut by a line parallel to the y-axis in two and only two points P_{1}, P_{2}. Let MP_{2 }> MP_{1}. Let AC and BD be two tangents at A and B respectively which are parallel to the y-axis. Let OC = a, OD = b.
If S is the area of the region AP_{1} BP_{2}, then S = S_{1} – S_{2}, where
S_{1} = Area of the region CDBP_{2}A
And S_{2} = Area of the region CDBP_{1}A.
Suppose that as t increases from t_{1} to t_{2}, the points (x, y) moving from an arbitrary point E on the curve returns to the point E when taken in counter-clockwise direction. Since the curve is closed, the point on it which corresponds to t = t_{1} is the same as the point which corresponds to t = t_{2}. Let the values of t corresponding to A and B be ta and tb, respectively.
It follows, from (1) and (2),
Similarly, by drawing the tangents parallel to the x-axis and applying the formula ∫x dy for the area bounded by a curve, the y-axis and to abscissa, we can show that
Adding (3) and (4) and dividing by 2, we obtain
This is often written as
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