Find a continuous function f, where (x4−4x2)≤f(x)≤(2x2−x3) such that the area bounded by y=f(x),y=x4−4x2, the y-axis, and the line x=t, where (0≤t≤2) is k times the area bounded by y=f(x),y=2x2−x3, y-axis, and line x=t(where 0≤t≤2).
The area of the region bounded by the parabola (y−2)2=x−1, the tangent to the parabola at the point (2,3), and the x-axis is
If y=f(x) is a monotonic function in (a,b), then the area bounded by the ordinates at x=a,x=b,y=f(x)andy=f(c)(where c∈(a,b)) is minimum when c=2a+b.
Proof : A=a∫c(f(c)−f(x))dx+c∫b(f(c))dx
Differentiating w.r.t. c, we get
For maxima and minima , dcdA=0
Also for c<2a+b,dcdA<0 and for c>2a+b,dcdA>0
Hence, A is minimum when c=2a+b.
If the area enclosed by f(x)=sinx+cosx,y=a between two consecutive points of extremum is minimum, then the value of a is
The value of the parameter a such that the area bounded by y=a2x2+ax+1, coordinate axes, and the line x=1 attains its least value is equal to