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=sin⁡x,    0≤x≤π2

That means,

a=0, b=π2

Substitute values of  a & b in the above inequality.

π2-0nfπ2-f0≤0.0001

π2nsinπ2-sin⁡0≤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