Consider two curves C1:y=x1andC2:y= In x on the xy plane. Let D1 denotes the region surrounded by C1,C2, and the line x=1 and D2 denotes the region surrounded by C1,C2 and the line x=a. If D1=D2, then the sum of logarithm of possible values of a 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
If Ai is the area bounded by ∣x−ai∣+∣y∣=bi,I∈N, where ai+1=ai+23biandbi+1=2bi,a1=0,=b1=32, then
Which of the following is the possible value/values of c for which the area of the figure bounded by the curves y=sin2x, the straight lines x=π/6,x=c and the abscissa axis is equal to 1/2?
Find the area bounded by the curve f(x)=x+sinxand its inverse function between the ordinates x=0 to x=2π.