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