Problem 3 - Single Variable Calculus

Question

You are given a function \(f\), an interval \([a, b]\), the number \(n\) of subintervals into which \([a, b]\) is divided \((\) each of length \(\Delta x=(b-a) / n)\), and the point \(c_{k}\) in \(\left[x_{k-1}, x_{k}\right]\), where \(1 \leq k \leq n .\) (a) Sketch the graph of f and the rectangles with base on \(\left[x_{k-1}, x_{k}\right]\) and height \(f\left(c_{k}\right)\), and (b) find the approximation \(\sum_{k=1}^{n} f\left(c_{k}\right) \Delta x\) of the area of the region \(S\) under the graph of \(f\) on \([a, b] .\) $$ f(x)=2 x+3, \quad[0,4], \quad n=5, \quad c_{k} \text { is the right endpoint } $$

Step-by-Step Solution

Answer

The Riemann sum approximation of the area under the curve \(f(x) = 2x + 3\) on the interval \([0, 4]\) using the right endpoints and 5 subintervals is \(31.2\).

Step 1: Calculate the length of each subinterval

Determine the length of each subinterval by dividing the interval \([a, b]\) into \(n\) equal parts using the formula \(\Delta x = \frac{b-a}{n}\). In this case, the interval is \([0, 4]\), and \(n = 5\): $$ \Delta x = \frac{4 - 0}{5} = \frac{4}{5} = 0.8 $$

Step 2: Determine the right endpoints of each subinterval

Since we are given that \(c_k\) is the right endpoint for each subinterval, we need to find the values of \(c_k\) for each \(k\) in the range \(1 \leq k \leq n\). In this case, we have: $$ c_1 = 0.8 \times 1 = 0.8 \\ c_2 = 0.8 \times 2 = 1.6 \\ c_3 = 0.8 \times 3 = 2.4 \\ c_4 = 0.8 \times 4 = 3.2 \\ c_5 = 0.8 \times 5 = 4 $$

Step 3: Sketch the graph of the function f and the rectangles

Plot the function \(f(x) = 2x + 3\) and the rectangles with base on \([x_{k-1}, x_k]\) and height \(f\left(c_k\right)\). For each subinterval, draw a vertical line segment at \(x = c_k\), and draw a rectangle with the height equal to the function value at that point \(f(c_k)\). The rectangles should show the approximation of the area under the curve.

Step 4: Calculate the Riemann sum approximation of the area

Use the Riemann sums formula \(\sum_{k=1}^{n} f\left(c_{k}\right) \Delta x\) to find the approximation of the area under the curve. In this case, we have: $$ \sum_{k=1}^{5} f\left(c_{k}\right) \Delta x = \sum_{k=1}^{5} (2c_k + 3) \cdot 0.8 $$ Substitute the values of \(c_k\) and calculate the sum: $$ (2(0.8) + 3)(0.8) + (2(1.6) + 3)(0.8) + (2(2.4) + 3)(0.8) + (2(3.2) + 3)(0.8) + (2(4) + 3)(0.8) $$ Calculate the areas of the rectangles and sum them up: $$ (4.6)(0.8) + (6.2)(0.8) + (7.8)(0.8) + (9.4)(0.8) + (11)(0.8) = 3.68 + 4.96 + 6.24 + 7.52 + 8.8 $$ Finally, add the sum of the areas of the rectangles: $$ 3.68 + 4.96 + 6.24 + 7.52 + 8.8 = 31.2 $$ The Riemann sum approximation of the area under the curve is \(31.2\).

Key Concepts

Subintervals

To approximate the area under a curve using Riemann sums, one of the initial steps involves dividing the interval \( [a, b] \) into smaller sections known as subintervals. These subintervals help in creating slices of the region under the curve by which we can estimate the total area. Each subinterval has an equal length \( \Delta x \), calculated as \( \Delta x = \frac{b-a}{n} \), where \( n \) is the number of subintervals.
In our given exercise, the interval \( [0, 4] \) is divided into 5 subintervals, thus each subinterval is of length \( \Delta x = 0.8 \). Working with subintervals allows us to piece together the overall area under a function graph by considering the contributions from each small section. This strategy simplifies complex areas by breaking them into manageable parts. It also lays the foundation for finding definite integrals, where these subintervals become infinitesimally small.

Function Graph

The function graph provides a visual representation of the mathematical function and is crucial in understanding how to approximate areas under it. For our example with \( f(x) = 2x + 3 \), the graph is a straight line with a slope of 2 and a \( y \)-intercept at 3.
Graphing this function involves:

• Starting at the point (0, 3) when \( x = 0 \).
• Drawing a line at a slope of 2, meaning for every 1 unit increase in \( x \), \( y \) increases by 2.

When approximating the area under this line, we not only focus on the curve but also visualize how rectangles can fit under it. These rectangles, based on specific points along our subintervals, closely represent segments of the area's total space beneath the graph between points 0 and 4. Understanding the graph is key to successfully constructing the right approximations and verifying the placement of rectangles during the area approximation process.

Area Approximation

Area approximation using Riemann sums involves summing up the areas of rectangles that are drawn under the function graph over the specified subintervals. Each rectangle corresponds to a subinterval and uses a sample point \( c_k \) within that subinterval. In our exercise, the right endpoints are used for calculation. The height of each rectangle is determined by \( f(c_k) \) — the function value at the chosen point.
Here's the formula used for approximation: \[ \sum_{k=1}^{n} f(c_k) \Delta x \]

• In this exercise, \( n = 5 \) subintervals.
• Each rectangle's area is \( f(c_k) \cdot \Delta x \).

By calculating the area of these rectangles and adding them together, we form a numerical estimation of the total area under the curve from 0 to 4. This method gained its name since the approximations get increasingly accurate as the number of subintervals approaches infinity.

Right Endpoint Method

In the Right Endpoint Method for Riemann sums, the value of \( c_k \) used for calculating the height of rectangles is taken as the right end of each subinterval. This method is useful because it often provides a better approximation than the left endpoint method when dealing with functions that are increasing over the interval.
Going back to our function \( f(x) = 2x + 3 \), for each subinterval \( [x_{k-1}, x_k] \), the right endpoint \( x_k \) becomes \( c_k \). Here are specific steps to apply this method:

• Determine each right endpoint: \( c_1, c_2, ..., c_5 \).
• For each \( c_k \), compute the function's value: \( f(c_k) \).
• Multiply by the width \( \Delta x \) to find the rectangle's area.

The total estimated area is simply the sum of all rectangle areas. Although this method can sometimes yield overestimates or underestimates, it is often straightforward for visualization and practical computation when paired with a consistent function like a straight line.