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 ___.
Let f:[0,∞)→R be a continuous function such that f(x)=1−2x+0∫xex−tf(t)dt for all x∈[0,∞). Then, which of the following statements(s) is (are)) TRUE?
Let Ar be the area of the region bounded between the curves y2=(e−kr)x(where k>0,r∈N) and the line y=mx(where m=0), k and m are some constants
n→∞limi=1ΣnAi=48(e2k−1)1 then the value of m 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
For a point P in the plane, let d1(P)andd2(P) be the distances of the point P from the lines x-y=0 and x+y=0 respectively. The area of the region R consisting of all points P lying in the first quadrant of the plane and satisfying 2≤d1(P)+d2(P)≤4, is