Estimate the area under the graph of f(x) - 1/x from x=1 to x=2 using four approximating rectangle and right endpoints. Is your estimate an underestimate or an overestimate?

To estimate the area under the graph of \( f(x) = \frac{1}{x} \) from \( x = 1 \) to \( x = 2 \) using four approximating rectangles and the right endpoints, follow these steps:

### Step 1: Determine the width of each subinterval
The interval from \( x = 1 \) to \( x = 2 \) has a total length of \( 2 - 1 = 1 \). Dividing this interval into 4 equal subintervals gives each rectangle a width of:

\[
\Delta x = \frac{2 - 1}{4} = 0.25
\]

### Step 2: Determine the right endpoints
Since we're using right endpoints, we need the \( x \)-coordinates of the right ends of each subinterval. The subintervals are:

- From \( x = 1 \) to \( x = 1.25 \)
- From \( x = 1.25 \) to \( x = 1.5 \)
- From \( x = 1.5 \) to \( x = 1.75 \)
- From \( x = 1.75 \) to \( x = 2 \)

The right endpoints are: \( x_1 = 1.25 \), \( x_2 = 1.5 \), \( x_3 = 1.75 \), and \( x_4 = 2 \).

### Step 3: Evaluate the function at the right endpoints
Now, calculate the value of the function \( f(x) = \frac{1}{x} \) at each right endpoint:

- \( f(1.25) = \frac{1}{1.25} = 0.8 \)
- \( f(1.5) = \frac{1}{1.5} \approx 0.6667 \)
- \( f(1.75) = \frac{1}{1.75} \approx 0.5714 \)
- \( f(2) = \frac{1}{2} = 0.5 \)

### Step 4: Calculate the area of each rectangle
The area of each rectangle is given by \( \text{Area} = f(x_i) \times \Delta x \), where \( f(x_i) \) is the value of the function at the right endpoint, and \( \Delta x = 0.25 \).

- Area of the first rectangle: \( 0.8 \times 0.25 = 0.2 \)
- Area of the second rectangle: \( 0.6667 \times 0.25 \approx 0.1667 \)
- Area of the third rectangle: \( 0.5714 \times 0.25 \approx 0.1429 \)
- Area of the fourth rectangle: \( 0.5 \times 0.25 = 0.125 \)

### Step 5: Add up the areas
Now, sum the areas of the four rectangles to get the total estimated area:

\[
\text{Total Area} = 0.2 + 0.1667 + 0.1429 + 0.125 \approx 0.6346
\]

### Step 6: Determine if it's an overestimate or an underestimate
Since we are using the right endpoints and the function \( f(x) = \frac{1}{x} \) is decreasing on the interval from \( x = 1 \) to \( x = 2 \), the rectangles overestimate the area under the curve. This is because the value of the function at the right endpoint is smaller than at the left endpoint for each subinterval.

### Final Answer:
The estimate of the area under the curve is approximately \( 0.6346 \), and it is an **overestimate**.

Would you like further details on any part of this solution?

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Here are five related questions to expand your understanding:
1. How would the estimate change if we used left endpoints instead of right endpoints?
2. What is the exact area under the curve between \( x = 1 \) and \( x = 2 \) using integration?
3. How would using more subintervals affect the accuracy of the estimate?
4. Can we use the midpoint rule to estimate the area and how does it compare?
5. What is the trapezoidal rule and how would it apply to this problem?

**Tip**: Increasing the number of subintervals in Riemann sums generally improves the accuracy of the area estimation.

Estimate the area under the graph of f(x) = 1/x from x = 1 to x = 2 using four approximating rectangles and right endpoints. Is your estimate an underestimate or an overestimate?

Estimate the area under the curve of f(x) = 1/x from x = 1 to x = 2 using four rectangles with right endpoints. This problem demonstrates Riemann sums, a key calculus concept, and shows how to determine if the estimate is an overestimate or an underestimate. Suitable for Grades 11-12.

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