To determine
To find:
A value of n
Answer
15708
Explanation
1) Concept:
The area A of the region S that lies under the graph of the continuous function f is the limit of the sum of the areas of approximating rectangles.
A=limn→∞Rn=limn→∞fx1∆x+fx2∆x+…+fxn∆x=limn→∞∑i=1nfxi∆x
The width of the interval a, b is (b-a), so the width of each n strip is
∆x=b-an
where x0=a and xn=b . The right end points of the subintervals are xi=a+i∆x.
2) Given:
i) Rn-A<0.0001
ii)
Rn-A<b-anfb-fa
3) Calculation:
Since
Rn-A<b-anfb-fa
Comparing this with Rn-A<0.0001, we get
b-anfb-fa≤0.0001
Solve this inequality for n.
It is also given that
y=sinx, 0≤x≤π2
That means,
a=0, b=π2
Substitute values of a & b in the above inequality.
π2-0nfπ2-f0≤0.0001
π2nsinπ2-sin0≤0.0001
π2n1-0≤0.0001
π2n1≤10-4
π2n≤10-4
π2·104≤n
1.5707963·104≤n
15707.963≤n
Conclusion:
n=15708