Estimate the area under the graph of $f(x)=\frac{1}{x+4}$ over the interval $[1,4]$ using twelve approximating rectangles and right endpoints. \[ R_{n}= \] Repeat the approximation using left endpoints. \[ L_{n}= \]
Solution
Step 1 ：Define the interval from a = 1 to b = 4 and the number of rectangles n = 12.
Step 2 ：Calculate the width of each rectangle dx = (b - a) / n = 0.25.
Step 3 ：Define the x-values at the right endpoints of the rectangles: x_right = [1.25, 1.5, 1.75, 2, 2.25, 2.5, 2.75, 3, 3.25, 3.5, 3.75, 4].
Step 4 ：Define the x-values at the left endpoints of the rectangles: x_left = [1, 1.25, 1.5, 1.75, 2, 2.25, 2.5, 2.75, 3, 3.25, 3.5, 3.75].
Step 5 ：Calculate the area under the curve using right endpoints: \(R_{n} = \sum_{i=1}^{n} f(x_{i}) \cdot dx\), where \(f(x) = \frac{1}{x+4}\). The result is \(R_{n} = 0.461\).
Step 6 ：Calculate the area under the curve using left endpoints: \(L_{n} = \sum_{i=1}^{n} f(x_{i}) \cdot dx\), where \(f(x) = \frac{1}{x+4}\). The result is \(L_{n} = 0.480\).
Step 7 ：Final Answer: The area under the graph of \(f(x)=\frac{1}{x+4}\) over the interval [1,4] using twelve approximating rectangles and right endpoints is approximately \(\boxed{0.461}\) and using left endpoints is approximately \(\boxed{0.480}\).
