With a programmable calculator or computer, it is possible to evaluate the expressions for the sums of areas of approximating rectangles, even for large values of $n$, using looping.

(a) Compute the sum of the areas of approximating rectangles for the region under $y = \sin(x)$ from $x = 0$ to $x = \pi$ by using equal subintervals and right endpoints for $n = 10, 30, 50$. Include at least six decimal places in your answers.

$R_{10} =$

$R_{30} =$

$R_{50} =$

(b) Based on your answers to (a), make an educated guess for the value of the exact area under $y = \sin(x)$ from $x = 0$ to $x = \pi.$

Exact area =