Area Under Cartesian Curve
The area of the region bounded by a curve y = ƒ(x), the axis of x, and two ordinates x = a and x = b is
∫^{b}_{a} ydx.
Proof: Let P (x, y), Q (x + Δx, y + Δy) be two neighbouring points on the given curve such that the functions y = ƒ(x) is increasing (or decreasing) in the interval [x, x + Δx]. Draw the ordinates PL and QM. Complete the rectangles MP and LQ. Suppose that
S = Area of the region CLPA (S is a function of x)
And S + ΔS = Area of the region CMQA.
∴ ΔS = Area of the region LMQP.
We have LM = OM – OL = x + Δx – x = Δx.
Clearly, Area of the rectangle MP < ΔS < Area of the rectangle LQ
y Δx < ΔS < (y + Δy) Δx
y < ΔS/Δx < y + Δy. (1)
Let QP so that Δx0. Then from (1) we obtain
dS/dx = y.
∴ ∫^{b}_{a} ydx = ∫^{b}_{a} dS/dx dx = |S|^{b}_{a}
= [S]x = b – [S]x = a
= Area of the region ABDC – 0.
Hence the required area = ∫^{b}_{a} ydx.
Remark: Similarly it can be proved that:
The area of the region bounded by a curve x = Ø(y), the axis of y, and the two lines y = c and y = d is
∫^{d}_{c} xdy.
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