Step 1 :Divide the interval [2,7] into 8 equal parts, so the width of each rectangle, \(\Delta x\), is \((7-2)/8 = 0.625\).
Step 2 :Calculate the right endpoint approximation (R_n):
Step 3 :The right endpoints of the intervals are: 2.625, 3.25, 3.875, 4.5, 5.125, 5.75, 6.375, 7.
Step 4 :\(R_n = \Delta x * [f(2.625) + f(3.25) + f(3.875) + f(4.5) + f(5.125) + f(5.75) + f(6.375) + f(7)]\)
Step 5 :\(= 0.625 * [1/(2.625+2) + 1/(3.25+2) + 1/(3.875+2) + 1/(4.5+2) + 1/(5.125+2) + 1/(5.75+2) + 1/(6.375+2) + 1/(7+2)]\)
Step 6 :\(= 0.625 * [0.2 + 0.1667 + 0.1429 + 0.125 + 0.1111 + 0.1 + 0.0909 + 0.0833]\)
Step 7 :\(= 0.625 * 1.0209 = 0.6381\) (rounded to 4 decimal places)
Step 8 :Calculate the left endpoint approximation (L_n):
Step 9 :The left endpoints of the intervals are: 2, 2.625, 3.25, 3.875, 4.5, 5.125, 5.75, 6.375.
Step 10 :\(L_n = \Delta x * [f(2) + f(2.625) + f(3.25) + f(3.875) + f(4.5) + f(5.125) + f(5.75) + f(6.375)]\)
Step 11 :\(= 0.625 * [1/(2+2) + 1/(2.625+2) + 1/(3.25+2) + 1/(3.875+2) + 1/(4.5+2) + 1/(5.125+2) + 1/(5.75+2) + 1/(6.375+2)]\)
Step 12 :\(= 0.625 * [0.25 + 0.2 + 0.1667 + 0.1429 + 0.125 + 0.1111 + 0.1 + 0.0909]\)
Step 13 :\(= 0.625 * 1.0866 = 0.6791\) (rounded to 4 decimal places)
Step 14 :So, the right endpoint approximation \(R_n\) is \(\boxed{0.6381}\) and the left endpoint approximation \(L_n\) is \(\boxed{0.6791}\).