Consider the function $f(x)=e^{-x^2},\;-2 \le x \le 2.$

(a) Estimate the area under the graph of $f$ using four approximating rectangles and taking the sample points to be right endpoints. Include at least six decimal places in your answers.

$R_{4} =$

(b) Repeat part (a), with midpoints as the sample points. Include at least six decimal places in your answers.

$M_{4} =$

(c) Now estimate the area under the graph of $f$ using eight approximating rectangles and taking the sample points to be right endpoints. Include at least six decimal places in your answers.

$R_{8} =$

(d) Repeat part (c), with midpoints as the sample points. Include at least six decimal places in your answers.

$M_{8} =$