Problem 17 - Calculus Early Transcendentals

Question

Use the figures to calculate the left and right Riemann sums for \(f\) on the given interval and for the given value of \(n\). $$f(x)=x+1 \text { on }[1,6] ; n=5$$

Step-by-Step Solution

Answer

Based on the provided step by step solution, we calculated the left Riemann sum as 20 and the right Riemann sum as 25 for the function f(x) = x+1 on the interval [1, 6], with n = 5 partitions.

Step 1: Determine the width of each partition

Since \([1,6]\) is the interval, the width of each partition is given by: $$\Delta x = \frac{b-a}{n} = \frac{6-1}{5} = 1$$

Step 2: Determine the partition endpoints

With the interval partitioned into 5 equal parts, we list the partition's endpoints: \(1, 2, 3, 4, 5, 6\).

Step 3: Calculate the left Riemann sum

The left Riemann sum is given by: $$L_n = \Delta x \sum_{i=0}^{n-1}f(x_{i})$$ Using our endpoints, we have $$L_n = 1[f(1)+f(2)+f(3)+f(4)+f(5)] = [f(1)+f(2)+f(3)+f(4)+f(5)]$$ Evaluate the function at these values: $$L_n = [(1+1)+(2+1)+(3+1)+(4+1)+(5+1)] = 2+3+4+5+6=20$$

Step 4: Calculate the right Riemann sum

The right Riemann sum is given by: $$R_n = \Delta x \sum_{i=1}^{n}f(x_{i})$$ Using our endpoints, we have $$R_n = 1[f(2)+f(3)+f(4)+f(5)+f(6)] = [f(2)+f(3)+f(4)+f(5)+f(6)]$$ Evaluate the function at these values: $$R_n = [(2+1)+(3+1)+(4+1)+(5+1)+(6+1)] = 3+4+5+6+7=25$$ The left Riemann sum for f(x) = x+1 on interval [1, 6] with n=5 partitions is 20, and the right Riemann sum is 25.

Key Concepts

Left Riemann Sum

Left Riemann sums are a method to approximate the area under a curve. This method uses the left endpoints of subintervals to estimate the area. To understand it with an example, consider the function \( f(x) = x + 1 \) on the interval \([1, 6]\). We want to divide this interval into 5 equal parts, which we call partitions.

The left Riemann sum can be calculated using the formula:

• \( L_n = \Delta x \sum_{i=0}^{n-1}f(x_{i}) \)

In this case, \( n = 5 \) and \( \Delta x = 1 \), the width of each partition. Using the left endpoints, which are \( 1, 2, 3, 4, \) and \( 5 \), you evaluate the function at these points:

• \( f(1) = 2 \)
• \( f(2) = 3 \)
• \( f(3) = 4 \)
• \( f(4) = 5 \)
• \( f(5) = 6 \)

Adding these values gives you the left Riemann sum of 20. This means the area approximated by the left Riemann sum is fewer rectangles underneath the curve, hence it may underestimate the integral if \( f(x) \) is increasing.

Right Riemann Sum

The right Riemann sum is similar to the left, but it uses the right endpoints of each subinterval instead. This method can often provide a different approximation of the area under the curve, depending on the function's behavior.

For the function \( f(x) = x + 1 \) on the interval \([1, 6]\) with 5 partitions, the right endpoints are \( 2, 3, 4, 5, \) and \( 6 \). The formula for the right Riemann sum is:

• \( R_n = \Delta x \sum_{i=1}^{n}f(x_{i}) \)

Here, \( \Delta x = 1 \), so you evaluate the function at the right endpoints:

• \( f(2) = 3 \)
• \( f(3) = 4 \)
• \( f(4) = 5 \)
• \( f(5) = 6 \)
• \( f(6) = 7 \)

The sum of these values gives a right Riemann sum of 25. For increasing functions, like the one in this problem, the right Riemann sum often results in a larger estimate, possibly overestimating the integral.

Partition Endpoints

In calculating both left and right Riemann sums, partition endpoints are crucial. These points define the subintervals over which the Riemann sums are calculated. For an interval \([a, b]\) divided into \( n \) equal parts, the partition endpoints include both the beginning and end points of each subinterval.

For our example, the interval \([1, 6]\) is divided into 5 partitions, giving us endpoints at these values: \( 1, 2, 3, 4, 5, \) and \( 6 \).

• \( 1 \) and \( 6 \) are the boundaries of the entire interval.
• \( 2, 3, \) and \( 4 \) are intermediate points that mark the ends of individual subintervals.

Deciding whether to use the left or right endpoints influences whether you're calculating a left or right Riemann sum. Knowing these endpoints helps determine where to evaluate the function and how to divide the area under the curve into rectangles.

Function Evaluation

Function evaluation is the process of determining the function's value at specific points. This is a key step in both left and right Riemann sums. Once you've determined your partition endpoints, you'll evaluate the function at each endpoint relevant to the method you're using.

For instance, with \( f(x) = x + 1 \), you calculate \( f(x) \) at each endpoint:

• \( f(1) = 2 \)
• \( f(2) = 3 \)
• \( f(3) = 4 \)
• \( f(4) = 5 \)
• \( f(5) = 6 \)
• \( f(6) = 7 \)

Evaluation provides each rectangle's height in the Riemann approximation; that is, \( f(x_i) \) gives how tall each rectangle is at \( x_i \). It's crucial for this step to be accurate because the sum of all these evaluations, multiplied by the width of each partition, gives the Riemann sums.