Question

Many computer algebra systems permit the evaluation of Riemann sums for left end point, right end point, or midpoint evaluations of the function. Using such a system, evaluate the 10 -subinterval Riemann sums using left end point, right end point, and midpoint evaluations. $$ \int_{0}^{1} \cos x d x $$

Short answer

Compute left, right, and midpoint Riemann sums using step equations.

Step-by-step solution

Define the Interval and Subintervals

The integral \( \int_{0}^{1} \cos x \, dx \) is over the interval \([0, 1]\). We will divide this interval into 10 equal parts. Each subinterval will therefore have a width of \( \Delta x = \frac{1-0}{10} = 0.1 \).

Left Endpoint Riemann Sum Evaluation

For the left endpoint Riemann sum, we evaluate the function \( \cos x \) at the left endpoint of each subinterval. Thus, the points used for evaluation will be \( x = 0, 0.1, 0.2, \ldots, 0.9 \). The Riemann sum is calculated as:\[S_L = \sum_{i=0}^{9} \cos(0.1i) \times 0.1\]

Right Endpoint Riemann Sum Evaluation

For the right endpoint Riemann sum, we evaluate the function \( \cos x \) at the right endpoint of each subinterval. Thus, the points used for evaluation will be \( x = 0.1, 0.2, \ldots, 1.0 \). The Riemann sum is then:\[S_R = \sum_{i=1}^{10} \cos(0.1i) \times 0.1\]

Midpoint Riemann Sum Evaluation

For the midpoint Riemann sum, we evaluate the function \( \cos x \) at the midpoint of each subinterval. Thus, the points used for evaluation will be \( x = 0.05, 0.15, \ldots, 0.95 \). The Riemann sum becomes:\[S_M = \sum_{i=0}^{9} \cos(0.05 + 0.1i) \times 0.1\]

Calculate Each Riemann Sum

Using a calculator or software, compute the values of \( S_L \), \( S_R \), and \( S_M \) from steps 2, 3, and 4. Make sure to use the appropriate function evaluations for each type of sum.

Summarize the Results

After computing each sum, summarize the results. Compare these approximations to the true value of the integral \( \int_{0}^{1} \cos x \, dx \), which is \( \sin(1) - \sin(0) \approx 0.8415 \).

Key concepts

Integral Approximation

Integral approximation is a technique used to estimate the value of a definite integral, especially when finding the exact value is complex or impossible.
Riemann sums are one of the methods used for this approximation, allowing us to estimate the area under a curve over a certain interval.
During this process, the interval is usually divided into smaller subintervals. Each of these subintervals contributes to the overall approximation by forming simple geometric shapes, like rectangles, that closely estimate the real area under the curve.Using Riemann sums, we can apply different approaches based on where within each subinterval we choose to evaluate our function. These approaches include the left endpoint, right endpoint, and midpoint methods.
Each method offers a slightly different estimate, and the goal is to find one close to the real integral value, which starts at 0 and ends at 1 in our exercise.To illustrate with the given function:

• The integral of \( \int_{0}^{1} \cos x \, dx \) represents the area under the cosine curve from the point 0 to 1.
• This task can be done using these different endpoint methods, offering various estimates of the area based on our chosen evaluation point in each subinterval.

Left Endpoint Method

The left endpoint method involves approximating the integral by evaluating the function at the left endpoints of the subintervals.
Each subinterval in the interval \([0, 1]\)is divided into equal parts. For our task, we use 10 subintervals, each with a width of \(\Delta x = 0.1\).For each subinterval, we approximate the area by assuming it forms a rectangle where:

• The height is determined by the function value at the left endpoint of each subinterval.
• The width is the constant subinterval width \(\Delta x = 0.1\).

The estimated integral using the left endpoint can be expressed as:

• \( S_L = \sum_{i=0}^{9} \cos(0.1i) \times 0.1 \).

By calculating this sum, each rectangle shaped by these parameters offers a part of the whole estimated area under the curve, adding to our complete approximation.

Right Endpoint Method

In contrast to the left endpoint method, the right endpoint method evaluates the function at the right endpoints of each subinterval.
This method often results in a different approximation because we shift our point of evaluation to the end of each subinterval.The interval \([0, 1]\)remains divided into 10 subintervals, but our chosen points for evaluation differ.
We calculate the area of rectangles where:

• The height of each rectangle is the value of the function at the "right" end of each subinterval, like \( x = 0.1, 0.2, \ldots, 1.0. \)
• The width is still \(\Delta x = 0.1\), consistent across all methods.

The right endpoint Riemann sum is then calculated as:

• \( S_R = \sum_{i=1}^{10} \cos(0.1i) \times 0.1 \).

This method, just like the left endpoint, gives an approximation but slightly differs in accuracy due to evaluating different reference points within each partition.

Midpoint Method

The midpoint method offers yet another way to approximate integrals by evaluating the function at the midpoint of each subinterval.
It is often preferred for its increased accuracy compared to the endpoint methods.As before, the interval \([0, 1]\)is divided into 10 equal subintervals of width \(\Delta x = 0.1\).Here's how the midpoint method works:

• Each subinterval's midpoint is calculated, such as \( x = 0.05, 0.15, \ldots, 0.95. \)
• These midpoints are used to determine the height of the rectangle approximating that portion of the integral.
• The width remains consistent at 0.1.

The Riemann sum using midpoints is formulated as:

• \( S_M = \sum_{i=0}^{9} \cos(0.05 + 0.1i) \times 0.1 \).

By using midpoints, the potential error due to approximation is generally reduced, providing a more reliable estimate of the actual integral value.