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
Consider the area S0,S1,S2…. bounded by the x-axis and half-waves of the curve y=e−xsinx, where x≥0.
The value of S0 is
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).
If a(a>0) is the value of parameter for each of which the are of the figure bounded by the straight line y=1+a4a2−ax and the parabola y=1+a4x2+2ax+3a2 is the greatest, then the value of a4 is ___
Let the straight line x= b divide the area enclosed by y=(1−x)2,y=0,andx=0 into two parts R1(0≤x≤b)andR2(b≤x≤1) such that R1−R2=41. Then b equals