Consider curves S1:∣x∣+∣y∣=a,S2:x2+y2=a2andS3: ∣x∣+∣y∣=a. If α is area bounded by S1andS2,β is area bounded by S1andS3andγ is the area bounded by S2andS3, then
Let C1andC2 be the graphs of the functions y=x2andy=2x, respectively, where 0≤x≤1. Let C3 be the graph of a function y=f(x), where 0≤x≤1,f(0)=0. For a point P on C1, let the lines through P, parallel to the axes, meet C2andC3 at Q and R, respectively (see figure). If for every position of P(onC1), the areas of the shaded regions OPQ and ORP are equal, determine the function f(x).
The ratio of the areas of two regions of the curve C1≡4x2+π2y2=4π2 divided by the curve C2≡y=−(sgn(x−2π))cosx (where sgn (x) = signum (x)) is
A curve y = f(x) is such that f(x)≥0andf(0)=0 and bounds a curvilinear triangle with the base [0,x] whose area is proportional to (n+1)th power of f(x)⋅ If f(1)=1 then find f(x).