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=

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=
Transcript text: Estimate the area under the graph of $f(x)=\frac{1}{x^{2}+2}$ over the interval $[3,6]$ using five approximating rectangles and right endpoints. \[ R_{n}= \] Repeat the approximation using left endpoints. \[ L_{n}= \]
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Solution

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Solution Steps

Step 1: Define the Interval and Number of Rectangles

We are estimating the area under the graph of f(x)=1x2+2 f(x) = \frac{1}{x^2 + 2} over the interval [3,6] [3, 6] using n=5 n = 5 rectangles.

Step 2: Calculate the Width of Each Rectangle

The width Δx \Delta x of each rectangle is calculated as: Δx=ban=635=0.6 \Delta x = \frac{b - a}{n} = \frac{6 - 3}{5} = 0.6

Step 3: Right Endpoint Approximation

Using the right endpoints, we evaluate the function at the points: x1=3.6,x2=4.2,x3=4.8,x4=5.4,x5=6.0 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 Rn R_n is given by: Rn=i=1nf(xi)Δx R_n = \sum_{i=1}^{n} f(x_i) \Delta x Calculating Rn R_n : Rn0.1297 R_n \approx 0.1297

Step 4: Left Endpoint Approximation

Using the left endpoints, we evaluate the function at the points: x0=3.0,x1=3.6,x2=4.2,x3=4.8,x4=5.4 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 Ln L_n is given by: Ln=i=0n1f(xi)Δx L_n = \sum_{i=0}^{n-1} f(x_i) \Delta x Calculating Ln L_n : Ln0.1684 L_n \approx 0.1684

Final Answer

The area under the graph using right endpoints is approximately Rn0.1297 R_n \approx 0.1297 and using left endpoints is approximately Ln0.1684 L_n \approx 0.1684 .

Thus, the final answers are: Rn0.1297 \boxed{R_n \approx 0.1297} Ln0.1684 \boxed{L_n \approx 0.1684}

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