#### 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*