0
You visited us 0 times! Enjoying our articles? Unlock Full Access!
Question

Let F(x)=x2+π6x2cos2tdt for all xϵR and f:[0,12][0,) be a continuous function. For αϵ[0,12], if F(a)+2 is the area of the region bounded by x=0,y=0,y=f(x) and x=a, then f(0) is

Solution
Verified by Toppr

F(x)=x2+π6x2cos2tdt
Differentiating both side w.r.t x
F(x)=2cos2(x2+π6)2x2cos2x
F(a)=4acos2(a2+π6)2cos2a
Now given, F(a)+2=a0f(x)dx
4acos2(a2+π6)2cos2a+2=a0f(x)dx
Differentiating both side w.r.t a
4cos2(a2+π6)8acos(a2+π6)sin(a2+π6)+4cosasina=f(a)
f(0)=4cos2(π6)=4×34=3

Was this answer helpful?
1
Similar Questions
Q1
Let F(x)=x2+π6x2cos2tdt for all xϵR and f:[0,12][0,) be a continuous function. For αϵ[0,12], if F(a)+2 is the area of the region bounded by x=0,y=0,y=f(x) and x=a, then f(0) is
View Solution
Q2
Let F(x)=x2+π6x2cos2t dt for all xR and f:[0,12][0,) be a continuous function. For a[0,12], if F(a)+2 is the area of the region bounded by x=0,y=0,y=f(x) and x=a, then f(0) is
View Solution
Q3
Let F(x)=x2+π6x2cos2t dt for all xR and f:[0,12][0,) be a continuous function. For a[0,12], if F(a)+2 is the area of the region bounded by x=0,y=0,y=f(x) and x=a, then f(0) is
View Solution
Q4
Let f:[0,)R be a continuous function such that
f(x)=12x+x0extf(t) dt for all x[0,). Then, which of the following statement(s) is (are) TRUE?
View Solution
Q5
Let f:[2,3][0,) be a continuous function such that f(1x)f(x) for all xϵ[2,3].
If R1 is the numerical value of the area of the region bounded by y=f(x),x=2,x=3 and the axis of x and R2=32xf(x)dx, then:-

View Solution