Some computer algebra systems have commands that will draw approximating rectangles and evaluate the sums of their areas, at least if $x_i^*$ is a left or right endpoint. For instance, in Maple use leftbox, rightbox, leftsum, and rightsum.

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(a) If $f(x) = \sin(\sin(x)),\quad 0 \le x \le \pi/2,$ find the left and right sums for $n = 10, 30, 50$. Include at least six decimal places in your answers.

$L_{10} =$

$L_{30} =$

$L_{50} =$

$R_{10} =$

$R_{30} =$

$R_{50} =$

(b) Using sketches of the approximating rectangles and your results in (a), the exact area under $f(x) = \sin(\sin(x)),\quad 1 \le x \le 4,$ is in which of the following intervals?

A. $[0.83,0.87]$
B. $[0.91,0.93]$
C. $[0.81,0.83]$
D. $[0.93,0.96]$
E. $[0.87,0.91]$