Problem 3 - Calculus: Early Transcendentals

Question

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

Step-by-Step Solution

Answer

Right: 0.635 (underestimate); Left: 0.76 (overestimate).

Step 1: Divide the Interval

We want to approximate the area under the curve from \( x = 1 \) to \( x = 2 \). First, we divide the interval \([1, 2]\) into four equal sub-intervals. The width of each rectangle, \( \Delta x \), will be \( (2-1)/4 = 0.25 \). This gives us the endpoints \( x_0 = 1 \), \( x_1 = 1.25 \), \( x_2 = 1.5 \), \( x_3 = 1.75 \), and \( x_4 = 2 \).

Step 2: Right Endpoint Approximation

For the right endpoint approximation, we will evaluate the function \( f(x) = \frac{1}{x} \) at the right endpoints \( x_1, x_2, x_3, \) and \( x_4 \). Calculate the heights: \( f(1.25) = \frac{1}{1.25}, f(1.5) = \frac{1}{1.5}, f(1.75) = \frac{1}{1.75}, f(2) = \frac{1}{2} \). The area of each rectangle is \( \Delta x \times f \text{(right endpoint)} \). Add these to estimate the total area: \( 0.25\left( \frac{1}{1.25} + \frac{1}{1.5} + \frac{1}{1.75} + \frac{1}{2} \right) \).

Step 3: Left Endpoint Approximation

For the left endpoint approximation, evaluate the function at the left endpoints \( x_0, x_1, x_2, \) and \( x_3 \). Calculate the heights: \( f(1) = \frac{1}{1}, f(1.25) = \frac{1}{1.25}, f(1.5) = \frac{1}{1.5}, f(1.75) = \frac{1}{1.75} \). The area of each rectangle is again \( \Delta x \times f \text{(left endpoint)} \). Sum these areas for the total: \( 0.25\left( 1 + \frac{1}{1.25} + \frac{1}{1.5} + \frac{1}{1.75} \right) \).

Step 4: Calculate Right Endpoint Areas

Calculate each value: \( f(1.25) = 0.8, f(1.5) = 0.67, f(1.75) = 0.57, f(2) = 0.5 \). Then the total area is \( 0.25 \times (0.8 + 0.67 + 0.57 + 0.5) = 0.25 \times 2.54 = 0.635 \).

Step 5: Calculate Left Endpoint Areas

Calculations give: \( f(1) = 1, f(1.25) = 0.8, f(1.5) = 0.67, f(1.75) = 0.57 \). The total area is \( 0.25 \times (1 + 0.8 + 0.67 + 0.57) = 0.25 \times 3.04 = 0.76 \).

Step 6: Determine Underestimate or Overestimate

The curve \( f(x) = \frac{1}{x} \) is decreasing over the interval \([1, 2]\). The right endpoint approximation is an underestimate since it does not capture the area beneath the curve after each right endpoint. The left endpoint is an overestimate because it includes extra area above the curve before reaching the next sub-interval.

Key Concepts

Right Endpoint Approximation

Right Endpoint Approximation is a technique used in integral calculus to estimate the area under a curve.

Left Endpoint Approximation

Left Endpoint Approximation is a method where the function value at the beginning of each sub-interval is used to estimate the area under a curve.

Rectangular Approximation Method

The Rectangular Approximation Method is a way of estimating the area under a curve by dividing it into smaller sections, essentially using rectangles to approximate the total area.