The area bounded by the curves y=cos x and y= sin x between the ordinates x=0 and x=3π/2 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 bounded by f(x)=3x3−x2+a and the straight lines x=0, x=2, and the x-axis is minimum, then the value of a is
A farmer F1 has a land in the shape of a triangle with vertices at P(0,0),Q(1,1)andR(2,0). From this land, a neighboring farmer F2 takes away the region which lies between the side PQ and curve of the from y=xn(n>1). If the area of the region taken away by the farmer F2 is exactly 30% of the area of ΔPQR, then the value of n is ___.
Area bounded by the relation [2x]+[y]=5,x,y>0 is__ (where [.] represents greatest integer funciton).
If S0,S1,S2,… are areas bounded by the x-axis and half-wave of the curve y=sinπx, then prove that S0,S1,S2,… are in A.P…
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