Area formulaWhen a function is in parametric formx= x(t)(α≤t≤β),y= y(t)Then the area of the region bounded by the curve and x=a.x=b isy(t)dx(t) = [" y(t)x(t)dt.ydx =Remark: in the formula, (x(α),y(α)is the left endpoint
Area formula ◼ When a function is in parametric form: Then the area of the region bounded by the curve and x=a, x=b is Remark: in the formula, is the left endpoint ( ( ), ( )) x y ( ) ( ), ( ) x x t t y y t = = ( ) ( ) ( ) ( ) . b a A ydx y t dx t y t x t dt = = =
ExampleEx. Find the area of the region bounded byb9Sol.x=acost, y=bsint~ ydx = 4/, bsin t(-a sin t)dt = πab.A=4|2
Example ◼ Ex. Find the area of the region bounded by ◼ Sol. 2 2 2 2 1. x y a b + = 0 0 2 4 4 sin ( sin ) . a = = − = A ydx b t a t dt ab x a t y b t = = cos , sin
Example Find the area under one arc of the cycloidx = a(t -sint), y = a(1-cost)" ydx = ("a(1 - cost)d[a(t -sint)]Sol. A== a2 [" (1 -cost) dt = 3元a2
Example ◼ Find the area under one arc of the cycloid ◼ Sol. x a t t y a t = − = − ( sin ), (1 cos ). 2 2 0 0 (1 cos ) [ ( sin )] a A ydx a t d a t t = = − − 2 2 2 2 0 a t dt a (1 cos ) 3 . = − =