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Please answer this calculus question?

Let F(x)=∫ sin(t^2) dt from 0 to x for 0≤x≤3. 
 
a) Use the Trapezoidal Rule with four equal subdivisions of the closed interval [0, 2] to approximate F(2).
b) On what interval or intervals is F increasing? Justify your answer.
c) If the average rate of change of F on the closed interval [0, 3] is k, find ∫ sin(t^2) dt from 0 to 3 in terms of k. 
 
Please show all your work.
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1 Answer

1) F(2)=∫02 sin(t2)dt
 
F(2)=0.5*[sin(0)+sin(0.52)]/2+0.5*[sin(0.52)+sin(1)]/2+0.5*[sin(1)+sin(1.52)]/2+0.5*[sin(1.52)+sin(22)]/2=0.5*[sin(0.52)+sin(1)+sin(1.52)+sin(4)/2]≈0.744
 
2) F(x) increases on the intervals where F'(x)>0. F'(x)=sin(x2). Since 0≤x≤3, F'(x)≥0 on the intervals [0,√π)]∪[√(2π);3] Therefore, those are the intervals where F(x) increases.
 
3) Average rate of change of F(x) is simply this: (1/3)∫03 sin(t2)dt. It is k, by the problem statement. So, the answer here is simply 3k.