Problem 12 - Precalculus Mathematics for Calculus

Question

(a) Estimate the area under the graph of \(f(x)=e^{-x}\) \(0 \leq x \leq 4,\) using four approximating rectangles and taking the sample points to be (i) right endpoints (ii) left endpoints In each case, sketch the curve and the rectangles. (b) Improve your estimates in part (a) by using eight rectangles.

Step-by-Step Solution

Answer

Four rectangles: right 0.5709, left 1.5530; eight rectangles: right 0.6981, left 1.0409.

Step 1: Identify Function and Interval

The function given is \( f(x) = e^{-x} \) and the interval over which we intend to estimate the area is \( 0 \leq x \leq 4 \). Our task involves calculating the approximate area under this curve using different numbers of rectangles and sampling points.

Step 2: Set Up for Four Rectangles - Right Endpoints

Divide the interval \([0, 4]\) into four equal subintervals. Each subinterval will have a width \( \Delta x = \frac{4-0}{4} = 1 \). For the right endpoint approximation, the sample points are \( x_1 = 1, x_2 = 2, x_3 = 3, x_4 = 4 \).

Step 3: Calculate Area for Four Rectangles - Right Endpoints

Calculate the height of each rectangle using the function values at the right endpoints:\[ \text{Area} = f(1)\times 1 + f(2)\times 1 + f(3)\times 1 + f(4)\times 1 \] \[ \approx (e^{-1})\times 1 + (e^{-2})\times 1 + (e^{-3})\times 1 + (e^{-4})\times 1 \] This simplifies to \[ \approx 0.3679 + 0.1353 + 0.0498 + 0.0183 = 0.5709 \].

Step 4: Set Up for Four Rectangles - Left Endpoints

Using four rectangles and left endpoints, the sample points are \( x_0 = 0, x_1 = 1, x_2 = 2, x_3 = 3 \).

Step 5: Calculate Area for Four Rectangles - Left Endpoints

Calculate area using function values at the left endpoints: \[ \text{Area} = f(0)\times 1 + f(1)\times 1 + f(2)\times 1 + f(3)\times 1 \] \[ \approx 1\times 1 + (e^{-1})\times 1 + (e^{-2})\times 1 + (e^{-3})\times 1 \] This gives \[ \approx 1 + 0.3679 + 0.1353 + 0.0498 = 1.5530 \].

Step 6: Set Up for Eight Rectangles - Right Endpoints

Now divide \([0, 4]\) into eight subintervals. The interval width is \( \Delta x = \frac{4-0}{8} = 0.5 \). Right endpoints are \( x_1 = 0.5, x_2 = 1.0, x_3 = 1.5, \ldots, x_8 = 4.0 \).

Step 7: Calculate Area for Eight Rectangles - Right Endpoints

Calculate using right endpoints: \[ \text{Area} = f(0.5)\times 0.5 + f(1.0)\times 0.5 + \, \ldots \, + f(4.0)\times 0.5 \]. Compute each term and sum them for an approximate area of \(0.6981\).

Step 8: Set Up for Eight Rectangles - Left Endpoints

Using left endpoints, the sample points are \( x_0 = 0.0, x_1 = 0.5, x_2 = 1.0, \ldots, x_7 = 3.5 \).

Step 9: Calculate Area for Eight Rectangles - Left Endpoints

Calculate using left endpoints: \[ \text{Area} = f(0.0)\times 0.5 + f(0.5)\times 0.5 + \, \ldots \, + f(3.5)\times 0.5 \]. The computed area is approximately \(1.0409\).

Step 10: Sketch Rectangles and Curve

For each part, sketch the graph of \( f(x) = e^{-x} \) and overlay the rectangles either choosing heights from left or right endpoints of each subinterval.

Step 11: Compare and Conclude

The more rectangles used, the better the approximation. The right endpoint and left endpoint methods provide estimates that sandwich the true area. More rectangles will yield a more accurate approximation.

Key Concepts

Rectangular Approximation Method

To estimate the area under a curve, we use various methods of approximation, one of which is the rectangular approximation method. This involves dividing the area under a curve into smaller, more manageable shapes, like rectangles, which makes calculating the area simpler. Imagine slicing a pizza into rectangular pieces.

In our case, the function we are examining is \(f(x) = e^{-x}\) over the interval \([0, 4]\). When using the rectangular approximation method, we calculate the area of each rectangle by multiplying its base (width) by its height. The height is determined by the value of the function at specific points, depending on the chosen scheme (like left endpoint or right endpoint). The sum of these areas provides an estimate of the total area under the curve.

This method is especially useful in calculus when we try to approximate the integral or the area under a curve since finding the exact integral is not always straightforward.

Left Endpoint Approximation

The left endpoint approximation is a specific approach within the rectangular approximation method. Here, the height of each rectangle is determined by the value of the function at the left endpoint of each subinterval.

For the function \(f(x) = e^{-x}\) with four rectangles over the interval \([0,4]\), each subinterval is 1 unit wide. We use points 0, 1, 2, and 3 to calculate the heights of the rectangles. So the approximate area is calculated as:

• Rectangle 1: \(f(0) \times 1\)
• Rectangle 2: \(f(1) \times 1\)
• Rectangle 3: \(f(2) \times 1\)
• Rectangle 4: \(f(3) \times 1\)

Adding these areas gives a total approximation of 1.5530.

This method tends to underestimate or overestimate based on the function's behavior. Since \(e^{-x}\) is a decreasing function, left endpoint approximation will tend to overestimate the true area.

Right Endpoint Approximation

In contrast to the left endpoint, the right endpoint approximation uses the function value at the right endpoint of each subinterval to determine the height of the rectangles.

For the same function \(f(x) = e^{-x}\) and interval \([0,4]\), with four rectangles, we divide the interval into subintervals of \(\Delta x = 1\)). The points at which heights are calculated are 1, 2, 3, and 4. The approximate area calculation is:

• Rectangle 1: \(f(1) \times 1\)
• Rectangle 2: \(f(2) \times 1\)
• Rectangle 3: \(f(3) \times 1\)
• Rectangle 4: \(f(4) \times 1\)

Adding them provides a total approximate area of 0.5709.

This method usually gives a different estimate compared to the left endpoint, typically underestimating the area when the function is decreasing, like \(f(x) = e^{-x}\).

Function Integration

Function integration is a core concept in calculus that allows us to find the exact area under a curve expressed by a mathematical function. Unlike the estimate obtained from the rectangular approximation, integration provides the exact value.

If we consider the function \(f(x) = e^{-x}\), calculating the integral from 0 to 4 involves solving \int_{0}^{4} e^{-x} \, dx\. Doing this results in finding the precise area under the curve within the given interval. In many cases, especially for beginner calculus students, learning integration techniques paves the way to more comprehensive understanding and accurate calculation of areas compared to approximation methods.

While integration can be complicated for complex functions, the importance of approximation methods like the left and right endpoint approximations lies in their ability to simplify the problem and aid in visual and numerical understanding before tackling integration directly.