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Estimate the area under the graph of $f(x)=\frac{1}{x+2}$ over the interval $[2,7]$ using eight approximating rectangles and right endpoints.
\[
R_{n}=
\]

Repeat the approximation using left endpoints.
\[
L_{n}=
\]

Report answers accurate to 4 places. Remember not to round too early in your calculations.
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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 \(\boxed{0.6791}\).