Questions: Estimate the area under the graph of f(x)=1/(x^2+2) over the interval [3,6] using five approximating rectangles and right endpoints. Rn= Repeat the approximation using left endpoints. Ln=

Solution Steps

We are estimating the area under the graph of $f(x) = \frac{1}{x^2 + 2}$ over the interval $[3, 6]$ using $n = 5$ rectangles.

The width $\Delta x$ of each rectangle is calculated as: $$\Delta x = \frac{b - a}{n} = \frac{6 - 3}{5} = 0.6$$

Using the right endpoints, we evaluate the function at the points: $$x_1 = 3.6, \quad x_2 = 4.2, \quad x_3 = 4.8, \quad x_4 = 5.4, \quad x_5 = 6.0$$ The Riemann sum $R_n$ is given by: $$R_n = \sum_{i=1}^{n} f(x_i) \Delta x$$ Calculating $R_n$: $$R_n \approx 0.1297$$

Using the left endpoints, we evaluate the function at the points: $$x_0 = 3.0, \quad x_1 = 3.6, \quad x_2 = 4.2, \quad x_3 = 4.8, \quad x_4 = 5.4$$ The Riemann sum $L_n$ is given by: $$L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x$$ Calculating $L_n$: $$L_n \approx 0.1684$$

Final Answer