Let O(0,0),A(2,0),andB(1,31 be the vertices of a triangle. Let R be the region consisting of all theose points P inside ΔOAB which satisfy d(P,OA)≤ min [d(P,AB)], where d denotes the distance from the point to the corresponding line. Sketch the region R and find its area.
Two curves C1≡[f(y)]2/3+[f(x)]1/3=0andC2≡[f(y)]2/3+[f(x)]2/3=12, satisfying the relation (x−y)f(x+y)−(x+y)f(x−y)=4xy(x2−y2)
The area bounded by the curve C2and∣x∣+∣y∣=12 is
Computing area with parametrically represented boundaries : If the boundary of a figure is represented by parametric equation, i.e., x=x(t),y=(t), then the area of the figure is evaluated by one of the three formulas :
Where αandβ are the values of the parameter t corresponding respectively to the beginning and the end of the traversal of the curve corresponding to increasing t.
The area of the region bounded by an are of the cycloid x=a(t−sint),y=a(1−cost) and the x-axis is